C₂²×#2243 #7623161

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

Aliases

direct product

C₂²×#2243

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

1846 conjugacy classes of subgroups; only the 658 normal subgroups are shown.

normal subgroups only

This group has 1846 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2);(3,4);(5,6)(7,10)(8,9)(11,12)(13,14)(15,24)(16,21)(17,20)(18,26)(19,23)(22,25)(27,28);(6,9)(10,12)(13,14,17,21,18,22,24,28,19,23,27,15,25,26,16,20);(5,7,8,11)(6,10,9,12)(13,15,16,23,25,28,27,22,19,21,24,14,18,20,17,26)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#33746, C₂²×#33873, C₂²×#33940
maximal normal subgroup of	C₂²×#33746, C₂²×#33873, C₂²×#33940

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530357);