#8105611

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

4628 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)(3,6)(4,5)(7,8)(9,10,15,18)(11,19,23,16)(12,13,17,14)(20,22,24,21)(25,26,29,31)(27,33,34,30)(28,36,38,32)(35,39,37,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,23)(10,16,18,19)(12,20,17,24)(13,21,14,22)(25,27,29,34)(26,30,31,33)(28,37,38,35)(32,40,36,39);(2,5)(3,8)(9,12)(10,13)(11,20)(14,18)(15,17)(16,21)(19,22)(23,24)(25,26)(27,30)(28,32)(29,31)(33,34)(35,39)(36,38)(37,40);(1,3,4,8)(2,6,5,7)(9,13,15,14)(10,12,18,17)(11,21,23,22)(16,20,19,24)(26,31)(27,34)(32,36)(35,37);(2,5)(6,7)(9,13,15,14)(10,17,18,12)(11,22,23,21)(16,20,19,24)(25,28,29,38)(26,32,31,36)(27,35,34,37)(30,39,33,40);(9,14,15,13)(10,12,18,17)(11,22,23,21)(16,20,19,24)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gh,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, 8012807);