#8699285

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

5213 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8606481);