C₂×#13691 #512925

order	512
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13691

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2263	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 464 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)(6,21)(7,26)(8,29)(9,33)(10,16)(11,18)(12,23)(13,28)(14,31)(17,46)(19,50)(20,45)(22,55)(24,54)(25,56)(27,36)(30,44)(32,62)(34,42)(35,59)(37,57)(38,63)(39,51)(40,58)(41,52)(43,49)(47,53)(48,66)(60,61)(64,65);(3,5,7,18,10,20,28,9)(4,11,13,15,16,33,26,45)(6,17,22,47,25,48,24,34)(8,19,27,49,32,52,60,31)(12,35,37,44,40,38,39,65)(14,36,41,29,43,61,50,62)(21,53,54,46,56,42,55,66)(23,30,51,59,58,64,57,63);(3,6)(4,12)(5,17)(7,22)(8,30)(9,34)(10,25)(11,35)(13,37)(14,42)(15,44)(16,40)(18,47)(19,51)(20,48)(21,29)(23,31)(24,28)(26,39)(27,59)(32,64)(33,38)(36,55)(41,66)(43,53)(45,65)(46,50)(49,58)(52,57)(54,61)(56,62)(60,63)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#513025, C₂×#513024, C₂×#512929
maximal normal subgroup of	C₂×#513025, C₂×#513024, C₂×#512929

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420121);