C₂×#13589 #512604

order	512
nilpotency class	5
isoclinism	isoclinic class of #1408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#13589

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
socle	C₂²	C₂×#353
prime radical	C₂×#3147	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1128 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1128 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);(3,4,7,12,8,13,23,30)(5,14,16,29,18,25,32,33)(6,10,19,27,20,11,22,28)(9,21,24,15,26,31,34,17);(3,5)(4,9)(6,15)(7,21)(8,18)(10,14)(11,25)(12,29)(13,26)(16,19)(17,20)(22,32)(23,31)(24,28)(27,34)(30,33);(3,6,8,20)(4,10,13,11)(5,15,18,17)(7,19,23,22)(9,14,26,25)(12,27,30,28)(16,31,32,21)(24,29,34,33)

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, 419800);