#511905

order	512
nilpotency class	4
isoclinism	isoclinic class of #511905
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32902
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33541	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1035 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419101);