#6485021

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6392217);