#8093050

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

4445 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,3,6,8)(2,7,4,5)(10,14)(15,20)(16,21)(22,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4,6,2)(3,7,8,5)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(2,4)(3,8)(10,14)(11,17)(16,21)(18,23)(26,32)(27,35)(31,36)(34,40);(1,5,6,7)(2,8,4,3)(10,14)(15,20)(16,21)(22,24);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=i,f^2=gh,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, 8000246);