C₂²×#2253 #7623208

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

Aliases

direct product

C₂²×#2253

Subgroups

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

1370 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 1370 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2);(3,4);(5,6,8,10)(7,11,12,9)(13,14,16,19)(15,23,25,37)(17,30,28,43)(18,31,29,22)(20,34,32,26)(21,36,33,24)(27,42,41,39)(35,40,38,44);(5,7,8,12)(6,9,10,11)(14,19)(20,32)(21,33)(22,31)(23,37)(30,43)(35,38)(39,42);(13,15,24,40,18,27,28,34,16,25,36,44,29,41,17,26)(14,20,30,39,22,35,33,37,19,32,43,42,31,38,21,23)

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, 7530404);