#8718933

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

4790 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)(3,8)(4,5)(6,7)(9,10,13,15)(11,14,16,12);(1,3,6,5)(2,4,7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,31)(30,32);(1,3,6,5)(2,4,7,8)(9,10)(11,12)(13,15)(14,16)(17,19,22,28)(18,23,26,27)(20,29,30,21)(24,25,32,31);(2,7)(3,5)(9,11,13,16)(10,14,15,12)(17,20,22,30)(18,24,26,32)(19,21,28,29)(23,31,27,25);(1,4,6,8)(2,3,7,5)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,31)(22,26)(25,29)(27,28)(30,32);(1,5,6,3)(2,4,7,8)(9,12,13,14)(10,11,15,16)(17,21)(18,25)(19,20)(22,29)(23,32)(24,27)(26,31)(28,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=i,e^2=g,f^2=h,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, 8626129);