#8182906

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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090102);