C₂²×#2236 #7623034

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂²×#2236

Subgroups

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

1850 conjugacy classes of subgroups; only the 658 normal subgroups are shown.

normal subgroups only

This group has 1850 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)(13,14,15,18)(16,19,22,26)(17,25,23,21)(20,28,27,24);(6,9)(10,12)(14,18)(19,26)(20,27)(21,25);(5,7,8,11)(6,10,9,12)(14,19)(16,24)(17,23)(18,26)(20,21)(22,28)(25,27)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530230);