#8093093

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

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, 8000289);