C₂×#12466 #490557

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#12466

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 424 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) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 66 points:

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

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, 397753);