C₂²×#2242 #7623159

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂²×#2242

Subgroups

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

1758 conjugacy classes of subgroups; only the 658 normal subgroups are shown.

normal subgroups only

This group has 1758 conjugacy classes of subgroups — too many to draw. Only its 658 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) = 256

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

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

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, 7530355);