#7644207

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

4636 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)(18,24)(19,20)(22,26)(28,31);(1,2)(3,7)(4,8)(5,6)(10,13)(14,16)(17,18)(19,25)(20,30)(21,26)(22,27)(23,24)(28,29)(31,32);(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,25,24,30)(21,31,27,28)(22,29,26,32);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,20,23,19)(18,25,24,30)(21,31,27,28)(22,32,26,29);(17,21,23,27)(18,26,24,22)(19,28,20,31)(25,29,30,32);(1,4,5,3)(2,8,6,7)(17,22,23,26)(18,27,24,21)(19,29,20,32)(25,28,30,31)

Polycyclic presentation:

2^9|c^2=hi,d^2=g,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, 7551403);