#483310

order	512
nilpotency class	4
isoclinism	isoclinic class of #483310
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1558
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

627 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,e^2=hi,f^2=gh,g^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fgh,g^b=gi,d^c=di,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390506);