#8750839

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

5172 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=gi,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, 8658035);