#8678463

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

5104 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)(3,7)(4,8)(5,6)(9,10,14,12)(11,15,13,16)(17,18)(19,28)(20,26)(21,23)(22,25)(24,27)(29,31)(30,32);(2,6)(3,8)(9,11,14,13)(10,15,12,16)(17,19,24,22)(18,25,27,28)(20,29,21,30)(23,31,26,32);(1,3)(2,4)(5,8)(6,7)(9,11,14,13)(10,16,12,15)(17,20,24,21)(18,26,27,23)(19,29,22,30)(25,32,28,31);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,15,13,16)(17,21,24,20)(18,26,27,23)(19,30,22,29)(25,32,28,31);(1,4)(2,3)(5,7)(6,8)(9,13,14,11)(10,15,12,16)(17,22,24,19)(18,25,27,28)(20,30,21,29)(23,31,26,32);(17,23,24,26)(18,21,27,20)(19,31,22,32)(25,30,28,29)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=gi,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, 8585659);