#72317791

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23900
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=gh,c^2=h,b^a=bg,d^a=df,d^b=dfhj,d^c=de,e^a=eh,e^b=eh,f^b=fhi,f^c=fh,g^d=ghi,g^f=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317812, #72317837, #72317914
maximal normal subgroup of	#72317812, #72317837, #72317914
maximal quotient of	#72318267, #72318311, #72338510

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