#427439

order	512
nilpotency class	3
isoclinism	isoclinic class of #427438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15369
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1189 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334635);