#512983

order	512
nilpotency class	5
isoclinism	isoclinic class of #512827
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	C₂⁴	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

375 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fg,g^a=gh,h^a=hi,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, 420179);