C₂×#13362 #498914

order	512
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#13362

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2065 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 2065 conjugacy classes of subgroups — too many to draw. Only its 194 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);(3,4,10,15)(5,16,20,42)(6,21,26,46)(7,27,31,13)(8,32,9,14)(11,28,38,18)(12,25,41,24)(17,49,51,65)(19,54,33,48)(22,47,55,66)(23,58,29,56)(30,44,34,45)(35,59,64,57)(36,52,37,53)(39,50,63,62)(40,60,43,61);(3,5)(4,11)(6,22)(7,28)(8,19)(9,33)(10,20)(12,39)(13,42)(14,36)(15,38)(16,27)(17,25)(18,31)(21,35)(23,59)(24,51)(26,55)(29,57)(30,53)(32,37)(34,52)(40,49)(41,63)(43,65)(44,54)(45,48)(46,64)(47,58)(50,60)(56,66)(61,62);(3,6,10,26)(4,12,15,41)(5,17,20,51)(7,29,31,23)(8,25,9,24)(11,35,38,64)(13,43,27,40)(14,46,32,21)(16,47,42,66)(18,50,28,62)(19,55,33,22)(30,60,34,61)(36,39,37,63)(44,56,45,58)(48,65,54,49)(52,59,53,57)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406110);