#8152208

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

4700 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)(3,5)(4,7)(6,8)(10,13)(14,16)(17,18,24,22)(19,27,23,25)(20,26,21,30)(28,32,29,31);(2,3)(6,7)(11,15)(14,16)(17,19,24,23)(18,25,22,27)(20,28,21,29)(26,31,30,32);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,24,21)(18,26,22,30)(19,28,23,29)(25,31,27,32);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,21,24,20)(18,26,22,30)(19,29,23,28)(25,31,27,32);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,22,24,18)(19,27,23,25)(20,30,21,26)(28,32,29,31);(17,23,24,19)(18,27,22,25)(20,29,21,28)(26,32,30,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8059404);