C₂×#12902 #494148

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12902

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	#1843
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 conjugacy classes of subgroups; only the 354 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 354 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);(3,4,7,12,8,13,23,34,9,14,24,35,25,36,49,60)(5,10,17,28,18,29,42,54,16,27,40,52,41,53,62,65)(6,11,21,32,22,33,48,59,20,31,46,57,47,58,64,66)(15,37,39,61,19,43,45,63,38,26,50,51,44,30,55,56);(3,5)(4,10)(6,19,20,44)(7,17)(8,18)(9,16)(11,30,31,43)(12,28)(13,29)(14,27)(15,22,38,47)(21,45,46,55)(23,42)(24,40)(25,41)(26,33,37,58)(32,56,57,63)(34,54)(35,52)(36,53)(39,48,50,64)(49,62)(51,59,61,66)(60,65);(3,6)(4,11)(5,15,16,38)(7,21)(8,22)(9,20)(10,26,27,37)(12,32)(13,33)(14,31)(17,39,40,50)(18,19,41,44)(23,48)(24,46)(25,47)(28,51,52,61)(29,30,53,43)(34,59)(35,57)(36,58)(42,45,62,55)(49,64)(54,56,65,63)(60,66)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#496097, C₂×#496723, C₂×#497049
maximal normal subgroup of	C₂×#496097, C₂×#496723, C₂×#497049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401344);