#8093089

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

4580 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:

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

Polycyclic presentation:

2^9|c^2=h,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, 8000285);