#483241

order	512
nilpotency class	4
isoclinism	isoclinic class of #483235
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1551
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60717	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

693 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=i,e^2=h,f^2=ghi,g^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390437);