#8658124

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

5009 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gh,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, 8565320);