C₂×#7646 #160940

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#7646

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1342 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 290 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) = 256

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68136);