C₂×#13724 #513034

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#13724

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

390 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 74
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 74 points:

(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,19,26)(13,21,31,51)(14,20,34,48)(15,23,37,44)(16,22,27,53)(17,40,42,69)(18,43,41,25)(24,56,57,35)(28,49,58,73)(29,62,64,74)(30,65,63,50)(32,45,66,70)(33,67,60,47)(36,68,38,55)(39,54,59,52)(46,71,72,61);(4,7)(8,10)(11,13,27,58,17,29,36,33,41,63,39,61,56,66,15,14,19,31,16,28,42,64,38,60,18,30,59,71,35,32,37,34)(12,20,44,70,24,46,52,50,43,67,55,62,69,73,22,21,26,48,23,45,57,72,54,65,25,47,68,74,40,49,53,51);(3,5,6,9)(4,8,7,10)(11,14,15,32,35,61,39,30,18,33,36,64,42,58,27,31,19,34,37,66,56,71,59,63,41,60,38,29,17,28,16,13)(12,21,22,49,40,62,55,47,25,50,52,72,57,70,44,48,26,51,53,73,69,74,68,67,43,65,54,46,24,45,23,20)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420230);