#72280276

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#3549
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂²	#60045
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=f,d^2=hij,e^2=hj,g^2=j,b^a=bh,c^a=chj,c^b=cfj,d^a=dg,d^b=dghi,e^a=eh,e^b=eghj,e^d=ej,f^a=fj,g^a=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280285
maximal normal subgroup of	#72280285

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