#7658438

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

4563 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565634);