C₂×#33982 #7625191

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

Aliases

direct product

C₂×#33982

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

1370 conjugacy classes of subgroups; only the 530 normal subgroups are shown.

normal subgroups only

This group has 1370 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)(5,11)(6,10)(7,24)(8,27)(9,15)(12,20)(13,21)(14,25)(16,32)(17,28)(18,46)(19,47)(22,39)(23,31)(26,53)(29,49)(30,48)(33,43)(34,44)(35,42)(36,52)(37,50)(38,51)(40,54)(41,64)(45,60)(55,65)(56,63)(57,61)(58,62)(59,66);(4,10)(6,20)(7,21)(8,25)(11,28)(12,13)(14,39)(15,31)(17,43)(18,44)(19,42)(22,52)(23,50)(24,29)(26,51)(27,30)(32,54)(33,34)(35,60)(36,53)(37,38)(40,61)(41,62)(45,66)(46,55)(47,56)(48,49)(57,58)(59,64)(63,65);(3,5)(4,11)(6,17)(7,18)(8,19)(9,16)(10,28)(12,33)(13,34)(14,35)(15,32)(20,43)(21,44)(22,45)(23,40)(24,46)(25,42)(26,41)(27,47)(29,55)(30,56)(31,54)(36,59)(37,57)(38,58)(39,60)(48,63)(49,65)(50,61)(51,62)(52,66)(53,64);(5,16)(11,32)(17,40)(18,41)(19,42)(28,54)(33,57)(34,58)(35,47)(43,61)(44,62)(45,63)(46,64)(55,59)(56,60)(65,66)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232007, C₂×#7625195, #72321377, …
maximal normal subgroup of	#72232007, C₂×#7625195, #72321377, …
maximal quotient of	#72322373, #72322427, #72322434

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532387);