C₂×#30238 #6435888

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#30238

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1144 conjugacy classes of subgroups; only the 490 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 490 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 — 5 generators on 66 points:

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

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, 6343084);