#8105497

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

4552 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:

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

Polycyclic presentation:

2^9|a^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, 8012693);