#8094479

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

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

5853 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)(6,8)(9,10,13,11)(12,15,16,14)(17,18,23,22)(19,26,29,24)(20,25,21,30)(27,32,28,31);(2,5)(3,7)(10,11)(14,15)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,31)(30,32);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,20,23,21)(18,25,22,30)(19,27,29,28)(24,31,26,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,21,23,20)(18,25,22,30)(19,28,29,27)(24,31,26,32);(17,22,23,18)(19,26,29,24)(20,25,21,30)(27,31,28,32);(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,31)(30,32)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8001675);