C₂²×#2249 #7623186

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂²×#2249

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂³×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2578 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 2578 conjugacy classes of subgroups — too many to draw. Only its 738 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 — 5 generators on 40 points:

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

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, 7530382);