#9223193

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

5363 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)(19,28)(20,25)(21,30)(22,26)(23,27)(24,31)(29,32);(1,2,4,6)(3,5,7,8)(9,10,12,15)(11,14,16,13)(17,19,23,31)(18,24,27,28)(20,29,21,26)(22,30,32,25);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,25,27,30)(19,29,31,26)(22,28,32,24);(9,10,12,15)(11,13,16,14)(17,21,23,20)(18,25,27,30)(19,26,31,29)(22,28,32,24);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,18)(19,24)(20,25)(21,30)(22,29)(23,27)(26,32)(28,31);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,22,23,32)(18,26,27,29)(19,30,31,25)(20,28,21,24)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,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
maximal normal subgroup of	#72280128
maximal quotient of	#72304113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9130389);