#512985

order	512
nilpotency class	5
isoclinism	isoclinic class of #512832
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#41	D₈
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

363 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420181);