C₂×#7065 #122326

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7065

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
socle	C₂³	#324
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

342 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 50
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 50 points:

(1,2);(3,4)(5,11,13,17)(6,14,15,9)(7,10)(8,12)(16,18)(19,20,22,27,23,28,33,41,24,26,32,38,34,39,45,48)(21,25,30,35,31,37,44,46,29,36,42,40,43,47,50,49);(3,5)(4,8,10,16)(6,12)(7,13)(9,11,14,17)(15,18)(19,21)(20,25,26,36)(22,30)(23,31)(24,29)(27,40,38,35)(28,37,39,47)(32,42)(33,44)(34,43)(41,49,48,46)(45,50)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29522);