C₂×#12171 #484649

order	512
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12171

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 872 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)(5,15)(6,20)(7,24)(8,30)(9,14)(10,17)(11,22)(12,29)(13,27)(16,43)(18,50)(19,42)(21,54)(23,36)(25,39)(26,59)(28,40)(31,37)(32,48)(33,49)(34,61)(35,62)(38,47)(41,45)(44,51)(46,64)(52,57)(53,56)(55,63)(58,66)(60,65);(3,5,9,19)(4,10,14,33)(6,16,22,45)(7,25,28,61)(8,31,27,18)(11,32,20,51)(12,37,38,50)(13,39,30,34)(15,40,42,24)(17,47,49,29)(21,55,26,60)(23,58,56,46)(35,64,52,66)(36,65,53,63)(41,54,43,59)(44,62,48,57);(3,6,9,22)(4,11)(5,16,19,45)(7,26,28,21)(8,23,27,56)(10,32)(12,35)(13,36)(14,20)(15,41)(17,48,49,44)(18,46,31,58)(24,59)(25,60,61,55)(29,62,47,57)(30,53)(33,51)(34,63)(37,64)(38,52)(39,65)(40,54)(42,43)(50,66)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391845);