C₂³×#490 #10575227

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂³×#490

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁶
prime radical	C₂³×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

10946 conjugacy classes of subgroups; only the 3746 normal subgroups are shown.

normal subgroups only

This group has 10946 conjugacy classes of subgroups — too many to draw. Only its 3746 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) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂³×#3180, C₂³×#3318, C₂³×#3366, …
maximal normal subgroup of	C₂³×#3180, C₂³×#3318, C₂³×#3366, …
maximal quotient of	C₂²×#33052, C₂²×#27264

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10482423);