#72308305

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂³×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂×#1999	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#399 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,c^2=j,d^2=g,e^2=ij,i^2=j,b^a=bg,c^a=cfgh,c^b=cfgj,d^a=dfg,d^b=df,d^c=dh,e^a=ehij,e^b=eh,e^c=eh,e^d=ehj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308428, #72308848
maximal normal subgroup of	#72308428, #72308848

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