C₂×#13590 #512605

order	512
nilpotency class	5
isoclinism	isoclinic class of #13584
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#13590

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	C₂×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1102 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419801);