C₂×#23805 #6432594

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#23805

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2914 conjugacy classes of subgroups; only the 722 normal subgroups are shown.

normal subgroups only

This group has 2914 conjugacy classes of subgroups — too many to draw. Only its 722 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) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 30 points:

(1,2);(7,8,10,11)(9,12,13,14)(15,16,19,24,21,23,20,25)(17,26,27,30,18,29,28,22);(3,4)(5,6)(8,11)(12,14)(15,17,21,18)(16,22,23,30)(19,28,20,27)(24,29,25,26);(4,6)(15,18,21,17)(16,22,23,30)(19,27,20,28)(24,29,25,26);(7,9,10,13)(8,12,11,14)(16,23)(22,30)(24,25)(26,29)

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, 6339790);