C₂×#32934 #6629001

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#32934

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4714 conjugacy classes of subgroups; only the 802 normal subgroups are shown.

normal subgroups only

This group has 4714 conjugacy classes of subgroups — too many to draw. Only its 802 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)(5,6)(7,8)(9,10)(11,12,15,17)(13,19,22,26)(14,23,20,18)(16,24,25,21);(11,13,20,24,15,22,14,21)(12,16,23,26,17,25,18,19);(4,6)(11,12)(13,16)(14,18)(15,17)(19,21)(20,23)(22,25)(24,26);(8,10)(12,17)(16,25)(18,23)(19,26)

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, 6536197);