#8664507

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

5059 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=g,e^2=g,f^2=g,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, 8571703);