#513015

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	#2215	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

347 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|c^2=i,d^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, 420211);