#513017

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	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

359 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 114 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)(9,10,12,18)(11,22,23,16)(13,17,27,39)(14,33,28,60)(15,34,29,21)(19,44,42,30)(20,46,43,31)(24,48,52,35)(25,58,53,63)(26,38,54,51)(32,47,59,41)(36,55,49,68)(37,56,50,69)(40,62,45,61)(57,65,70,67)(64,71,66,72);(2,5)(6,8)(9,11)(10,16)(12,23)(13,24)(14,25)(15,26)(17,35)(18,22)(19,36)(20,37)(21,38)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,63)(34,51)(39,48)(40,64)(41,65)(42,49)(43,50)(44,68)(45,66)(46,69)(47,67)(58,60)(59,70)(61,71)(62,72);(10,17)(13,30)(14,31)(15,29)(16,35)(18,39)(19,20)(21,47)(22,48)(24,55)(25,56)(26,54)(27,44)(28,46)(32,62)(33,40)(34,41)(36,37)(38,67)(42,43)(45,60)(49,50)(51,65)(52,68)(53,69)(57,72)(58,66)(59,61)(63,64)(70,71);(1,3,4,7)(2,6,5,8)(11,23)(16,22)(24,52)(25,53)(26,54)(35,48)(36,49)(37,50)(38,51)(55,68)(56,69)(57,70)(58,63)(64,66)(65,67)(71,72)

Polycyclic presentation:

2^9|a^2=i,d^2=ei,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, 420213);