C₂×#7335 #149520

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7335

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂³	C₂²⋊C₁₆
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

380 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 74
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 74 points:

(1,2);(3,4)(5,9,10,7)(6,8)(11,12,15,21,16,22,34,47,17,23,35,48,37,50,62,70,18,24,36,49,38,51,63,71,14,20,31,44,32,45,59,68)(13,25,27,52,28,53,56,72,26,42,54,46,55,66,73,69,29,43,33,60,57,67,61,74,30,19,39,40,58,41,64,65);(3,5)(4,7)(6,10)(8,9)(11,13)(12,19)(14,30)(15,33)(16,28)(17,26)(18,29)(20,43)(21,46)(22,41)(23,25)(24,42)(27,36)(31,54)(32,58)(34,61)(35,39)(37,55)(38,57)(40,49)(44,52)(45,67)(47,69)(48,60)(50,53)(51,66)(56,63)(59,73)(62,64)(65,71)(68,72)(70,74)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56716);