C₂×#13573 #512583

order	512
nilpotency class	5
isoclinism	isoclinic class of #13569
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13573

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1460 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1460 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);(3,4,6,11,8,13,21,31)(5,14,17,25,15,27,33,9)(7,12,23,10,24,28,20,29)(16,32,22,30,19,26,18,34);(3,5)(4,9)(6,19)(7,22)(8,15)(10,27)(11,30)(12,32)(13,25)(14,29)(16,21)(17,20)(18,24)(23,33)(26,28)(31,34);(5,15)(9,25)(14,27)(16,19)(17,33)(18,22)(26,32)(30,34)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#152676, #72279392
maximal normal subgroup of	C₂×#152676, #72279392
maximal quotient of	#72279392

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419779);