C₂×#32860 #6631552

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#32860

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1788 conjugacy classes of subgroups; only the 586 normal subgroups are shown.

normal subgroups only

This group has 1788 conjugacy classes of subgroups — too many to draw. Only its 586 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) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538748);