C₂²×#2245 #7623163

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂²×#2245

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

1674 conjugacy classes of subgroups; only the 658 normal subgroups are shown.

normal subgroups only

This group has 1674 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) = 512

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530359);