#72314124

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#16508
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂²×#575	C₂²
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

5827 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=h,d^2=ghj,e^2=j,b^a=beg,c^a=ci,c^b=cf,d^a=dghj,d^b=dghj,d^c=degh,e^b=ej,f^a=fj,g^c=gj,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314133, #72314177, #72314283
maximal normal subgroup of	#72314133, #72314177, #72314283

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.5);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);