#8093482

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

4466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,5)(6,8)(9,10)(11,12)(13,21)(14,15)(16,19)(17,22)(18,23)(20,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(2,5)(3,7)(10,15)(12,16)(17,21)(20,23)(26,30)(31,33)(32,37)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,12,15,16)(13,18,22,24)(17,20,21,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000678);