#8505661

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

4976 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)(3,5)(4,7)(6,8)(10,13)(14,16)(17,18)(19,25)(20,29)(21,22)(23,24)(26,32)(27,28)(30,31);(2,3)(6,7)(11,15)(14,16)(18,24)(21,32)(25,29)(27,30);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,25,24,29)(21,27,32,30)(22,28,26,31);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,23,19)(18,25,24,29)(21,30,32,27)(22,28,26,31);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,21)(18,26)(19,27)(20,30)(22,24)(23,32)(25,31)(28,29);(17,22,23,26)(18,21,24,32)(19,28,20,31)(25,27,29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412857);