C₂×#23840 #6432695

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#23840

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

1810 conjugacy classes of subgroups; only the 690 normal subgroups are shown.

normal subgroups only

This group has 1810 conjugacy classes of subgroups — too many to draw. Only its 690 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) = 34
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4,7,10)(5,9,6,8)(11,12,15,18)(13,17,14,16)(19,20,23,26)(21,24,28,32)(22,25,30,34)(27,31,29,33);(3,5,7,6)(4,8,10,9)(11,13,15,14)(12,16,18,17)(19,21)(20,24)(22,27)(23,28)(25,31)(26,32)(29,30)(33,34);(3,5,7,6)(4,9,10,8)(11,14,15,13)(12,17,18,16)(21,27)(22,30)(24,31)(25,34)(28,29)(32,33);(3,6,7,5)(4,8,10,9)(21,28)(24,32)(27,29)(31,33)

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, 6339891);