#8720349

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

5133 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,d^2=ghi,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, 8627545);