C₂×#7425 #150256

order	512
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7425

Subgroups

	subgroup	G/H
center, socle	C₂²	#1347
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×M₃₂	C₂³
prime radical	C₂⁴	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

299 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 66 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57452);