C₂×#33983 #7625192

order	512
nilpotency class	5
isoclinism	isoclinic class of #33982
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#33983

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁵
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1344 conjugacy classes of subgroups; only the 530 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 530 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 — 5 generators on 66 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#7625195
maximal normal subgroup of	C₂×#7625195
maximal quotient of	#72322380, #72322434, #72322440

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532388);