#72279251

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1390
commutator	C₂²×Q₈	C₂×C₄²
Frattini, cosocle	C₂²×#124	C₂³
prime radical	C₂⁴×D₄	D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂³×#124 ◁ G
LECS	G ▷ C₂²×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

3207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=d,c^2=ef,d^2=i,g^2=j,h^2=j,c^a=cg,c^b=ce,d^c=dh,f^a=fj,g^a=gh,g^d=gij,h^a=hi,h^c=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279281, #72306349, #72306356
maximal normal subgroup of	#72279281, #72306349, #72306356

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