#8527416

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8434612);