#8678929

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

4891 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8586125);