#8722935

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

5046 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gi,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, 8630131);