#7644311

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

4572 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)(19,26)(20,23)(21,29)(22,25)(24,30)(27,28)(31,32);(1,2)(3,7)(4,8)(5,6)(10,13)(14,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31);(9,10,12,13)(11,14,15,16)(19,23)(20,26)(27,31)(28,32);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30);(17,20,22,26)(18,23,25,19)(21,28,30,32)(24,31,29,27);(1,4,5,3)(2,8,6,7)(17,21,22,30)(18,24,25,29)(19,27,23,31)(20,28,26,32)

Polycyclic presentation:

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

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, 7551507);