C₂²×#2226 #7622996

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#2226

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

1870 conjugacy classes of subgroups; only the 698 normal subgroups are shown.

normal subgroups only

This group has 1870 conjugacy classes of subgroups — too many to draw. Only its 698 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);(5,6,8,10)(7,11,12,9)(13,14,17,21)(15,18,24,27)(16,25,23,20)(19,28,26,22);(5,7,8,12)(6,9,10,11);(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#33669, C₂²×#33912, C₂²×#33933, …
maximal normal subgroup of	C₂²×#33669, C₂²×#33912, C₂²×#33933, …
maximal quotient of	C₂×#512972, C₂²×#12937

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530192);