#8429374

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

5001 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8336570);