#9718192

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

4599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9625388);