#8105533

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4529 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:

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

Polycyclic presentation:

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

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, 8012729);