C₂×#13598 #512627

order	512
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13598

Subgroups

	subgroup	G/H
center	C₂×C₄	#354
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
socle	C₂²	C₂×#354
prime radical	C₂×#532	C₂²
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

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

normal subgroups only

This group has 832 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);(3,4,7,12,8,13,23,31)(5,14,17,32,16,25,33,30)(6,10,19,28,20,11,22,29)(9,24,26,18,27,21,34,15);(3,5,8,16)(4,9)(6,15,20,18)(7,21,23,24)(10,25)(11,14)(12,30)(13,27)(17,22,33,19)(26,29)(28,34)(31,32);(3,6,8,20)(4,10,13,11)(5,15,16,18)(7,19,23,22)(9,25,27,14)(12,28,31,29)(17,24,33,21)(26,30,34,32)

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, 419823);