C₂×#32816 #6631482

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

direct product

C₂×#32816

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×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 conjugacy classes of subgroups; only the 538 normal subgroups are shown.

normal subgroups only

This group has 1862 conjugacy classes of subgroups — too many to draw. Only its 538 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) = 50
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 50 points:

(1,2);(3,4,6,7)(5,8,9,10)(11,12,14,16)(13,15,17,18)(19,20)(21,25)(22,32)(23,27)(24,28)(26,34)(29,47)(30,40)(31,41)(33,50)(35,46)(36,44)(37,45)(38,43)(39,48)(42,49);(4,7)(5,9)(11,13)(12,15)(14,17)(16,18)(19,21,23,30)(20,25,27,40)(22,33,34,49)(24,37,36,31)(26,42,32,50)(28,45,44,41)(29,38,48,35)(39,46,47,43);(5,9)(8,10)(12,16)(15,18)(19,22,23,34)(20,26,27,32)(21,29,30,48)(24,38,36,35)(25,39,40,47)(28,46,44,43)(31,49,37,33)(41,50,45,42);(3,5,6,9)(4,8,7,10)(13,17)(15,18)(20,27)(21,30)(26,32)(28,44)(29,48)(31,37)(33,49)(43,46)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538678);