#8746609

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

5058 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653805);