#495790

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#392	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402986);