#7658992

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

4758 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=g,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, 7566188);