C₂×#12118 #484230

order	512
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12118

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1072 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1072 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)(5,17)(6,22)(7,26)(8,15)(9,14)(10,16)(11,19)(12,24)(13,29)(18,46)(20,54)(21,45)(23,57)(25,42)(27,35)(28,38)(30,43)(31,39)(32,41)(33,40)(34,61)(36,52)(37,53)(44,48)(47,55)(49,66)(50,64)(51,62)(56,60)(58,63)(59,65);(3,5,10,21)(4,11,16,37)(6,18,25,50)(7,27,30,52)(8,20,9,33)(12,34,39,63)(13,40,41,54)(14,36,15,35)(17,43,45,26)(19,32,53,29)(22,55,42,62)(23,58,28,61)(24,48,31,49)(38,64,57,46)(44,65,66,56)(47,59,51,60);(3,6)(4,12)(5,18)(7,28)(8,31)(9,24)(10,25)(11,34)(13,38)(14,42)(15,22)(16,39)(17,44)(19,51)(20,49)(21,50)(23,30)(26,56)(27,61)(29,59)(32,60)(33,48)(35,55)(36,62)(37,63)(40,64)(41,57)(43,65)(45,66)(46,54)(47,53)(52,58)

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, 391426);