#72250045

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24033
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#6443272	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(7,17,21,34)(8,23)(9,20)(10,25)(12,26,29,36)(13,14)(15,31)(18,32,35,22)(19,27,30,33)(24,28);(2,3)(7,18)(8,9)(10,25)(12,27)(13,24)(14,28)(15,31)(17,32)(19,36)(20,23)(21,35)(22,34)(26,30)(29,33);(1,3,4,2)(5,6)(7,19)(8,13)(9,24)(10,15)(11,16)(12,22)(14,23)(17,33)(18,36)(20,28)(21,30)(25,31)(26,35)(27,34)(29,32);(2,3)(5,7,20,34,10,22,9,18,11,21,8,17,25,32,23,35)(6,12,28,36,15,30,24,33,16,29,13,26,31,19,14,27)

Polycyclic presentation:

2^10|a^2=f,c^2=h,d^2=ehj,e^2=i,g^2=ij,i^2=j,b^a=bh,c^a=cfh,c^b=ch,d^a=de,d^b=dg,d^c=defgij,e^a=ei,e^b=eij,f^d=fj,g^a=gij,g^b=gi,g^c=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250103, #72256950, #72256967
maximal normal subgroup of	#72250103, #72256950, #72256967

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.5*F.8*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);