#9228933

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5293 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280721
maximal normal subgroup of	#72280721

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9136129);