#8093722

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

4744 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)(9,10,12,14)(11,13,16,15)(17,18)(19,22)(20,25)(21,31)(23,26)(24,29)(27,32)(28,30);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,30)(31,32);(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,20,23,21)(18,25,26,31)(19,27,29,28)(22,32,24,30);(9,11,12,16)(10,13,14,15)(17,21,23,20)(18,25,26,31)(19,28,29,27)(22,32,24,30);(1,2)(3,6)(4,5)(7,8)(17,22,23,24)(18,19,26,29)(20,30,21,32)(25,28,31,27);(1,3)(2,6)(4,7)(5,8)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000918);