C₂×#13721 #513030

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#13721

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂²×D₁₆	C₂²
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

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

normal subgroups only

This group has 404 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) = 256

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420226);