C₂×#13599 #512628

order	512
nilpotency class	5
isoclinism	isoclinic class of #13595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13599

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	C₂⁴	#99
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

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

normal subgroups only

This group has 816 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)(5,14,16,32,15,31,33,28)(7,12,22,29,23,10,19,27)(9,24,26,17,25,18,34,21);(3,5,8,15)(4,9)(6,18,20,24)(7,21,23,17)(10,14)(11,28)(12,31)(13,25)(16,19,33,22)(26,27)(29,34)(30,32);(5,15)(9,25)(14,31)(16,33)(17,21)(18,24)(26,34)(28,32)

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, 419824);