#8093122

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

5008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000318);