#495673

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#2388	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

999 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 999 conjugacy classes of subgroups — too many to draw. Only its 274 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402869);