#8093091

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4561 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:

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

Polycyclic presentation:

2^9|c^2=gh,e^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, 8000287);