#8709254

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

4785 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8616450);