C₂×#7327 #128167

order	512
nilpotency class	4
isoclinism	isoclinic class of #7307
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×#7327

Subgroups

	subgroup	G/H
center, socle	C₂³	#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

596 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 138 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)(5,11,12,8)(6,10)(7,9)(13,17,18,15)(14,16)(19,20,23,27,24,28,36,43)(21,29,31,44,32,45,35,49)(22,26,33,40,34,41,48,50)(25,37,38,30,39,46,42,47);(3,5,6,13,7,12,14,18)(4,8,10,15,9,11,16,17)(19,21)(20,25)(22,30,34,47)(23,35)(24,32)(26,29,41,45)(27,42)(28,39)(31,36)(33,37,48,46)(38,43)(40,49,50,44)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35363);