#8097283

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

4617 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:

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

Polycyclic presentation:

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

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, 8004479);