#513020

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	#33740	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 353 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) = 256

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

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

Polycyclic presentation:

2^9|c^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, 420216);