#8722154

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

4977 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8629350);