#9223196

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

5213 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gi,e^2=g,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	#72280128, #72283234
maximal normal subgroup of	#72280128, #72283234
maximal quotient of	#72304040

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9130392);