C₂×#12437 #490482

order	512
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12437

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
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

498 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 146 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)(5,19,25,49)(6,26,29,57)(7,15,31,43)(8,14,30,44)(9,17,10,16)(12,34,39,21)(13,40,42,27)(20,50,53,61)(22,38,54,37)(23,48,24,36)(28,58,33,45)(32,46,59,41)(35,52,62,56)(47,60,64,51)(55,65,66,63);(3,5,8,22,9,23,31,34,11,25,30,54,10,24,7,21)(4,12,15,36,16,37,44,49,18,39,43,48,17,38,14,19)(6,20,32,55,28,52,40,60,29,53,59,66,33,56,27,51)(13,35,45,63,41,61,57,64,42,62,58,65,46,50,26,47);(3,6)(4,13)(5,20)(7,27)(8,32)(9,28)(10,33)(11,29)(12,35)(14,26)(15,45)(16,41)(17,46)(18,42)(19,47)(21,51)(22,55)(23,52)(24,56)(25,53)(30,59)(31,40)(34,60)(36,63)(37,61)(38,50)(39,62)(43,58)(44,57)(48,65)(49,64)(54,66)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397678);