C₂×#7514 #153431

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7514

Subgroups

	subgroup	G/H
center	C₂×C₃₂	D₄
commutator	C₄	C₂²×C₃₂
Frattini, cosocle	C₂×C₃₂	C₂³
socle	C₂²	C₂²⋊C₃₂
prime radical	C₂×#41	C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

182 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 130
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 130 points:

(1,2);(3,4,7,14,8,15,30,49,9,16,31,50,34,53,77,98,10,17,32,51,35,54,78,99,37,56,80,101,82,103,118,126,11,18,33,52,36,55,79,100,38,57,81,102,83,104,119,127,6,13,26,45,27,46,71,93,28,47,72,94,73,95,114,124)(5,19,21,58,22,59,64,105,23,60,65,106,66,107,111,128,20,43,61,48,62,89,108,96,63,90,109,97,110,122,129,125,24,44,29,74,67,91,75,115,68,92,76,116,112,123,117,130,25,12,39,40,69,41,84,86,70,42,85,87,113,88,120,121);(3,5)(4,12)(6,25)(7,29)(8,22)(9,23)(10,20)(11,24)(13,44)(14,48)(15,41)(16,42)(17,19)(18,43)(21,33)(26,61)(27,69)(28,70)(30,75)(31,76)(32,39)(34,66)(35,62)(36,67)(37,63)(38,68)(40,52)(45,58)(46,91)(47,92)(49,96)(50,97)(51,74)(53,88)(54,59)(55,89)(56,60)(57,90)(64,79)(65,81)(71,108)(72,109)(73,113)(77,117)(78,84)(80,85)(82,110)(83,112)(86,100)(87,102)(93,105)(94,106)(95,123)(98,125)(99,115)(101,116)(103,107)(104,122)(111,119)(114,129)(118,120)(121,127)(124,128)(126,130)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60627);