C₂×#12889 #494082

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#12889

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

738 conjugacy classes of subgroups; only the 402 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 402 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) = 46
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 46 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401278);