#512946

order	512
nilpotency class	5
isoclinism	isoclinic class of #512941
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

442 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|b^2=eg,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gh,h^a=hi,d^b=di,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232017, #72322387, #72322434
maximal normal subgroup of	#72232017, #72322387, #72322434

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420142);