#513224

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
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	#33846	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

361 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420420);