#8105539

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

4419 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)(13,16)(14,18)(15,22)(20,23)(21,24)(25,26,31,34)(27,30,33,29)(28,36,38,32)(35,39,40,37);(1,3,5,7)(2,4,6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19)(25,27,31,33)(26,29,34,30)(28,35,38,40)(32,39,36,37);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,17)(13,16)(14,18)(19,22)(20,24)(21,23)(25,28)(26,32)(27,35)(29,39)(30,37)(31,38)(33,40)(34,36);(3,7)(4,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24)(25,29,31,30)(26,33,34,27)(28,37,38,39)(32,35,36,40);(2,6)(3,7)(9,12,14,13)(10,17,18,16)(11,21,22,20)(15,23,19,24)(25,30,31,29)(26,27,34,33)(28,37,38,39)(32,35,36,40);(9,13,14,12)(10,16,18,17)(11,21,22,20)(15,23,19,24)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,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, 8012735);