#6498272

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1258 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)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,32)(27,35,38,48)(28,33,39,46)(29,41,42,34)(31,43,44,37)(36,49,50,56)(40,53,52,47)(45,54,55,51);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,29,37,30,38,42,43)(26,31,34,35,32,44,41,48)(28,36,40,51,39,50,52,54)(33,45,47,49,46,55,53,56);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(26,32)(28,39)(31,44)(34,41)(35,48)(36,50)(40,52)(51,54);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,28,30,39)(26,33,32,46)(27,36,38,50)(29,40,42,52)(31,45,44,55)(34,47,41,53)(35,49,48,56)(37,51,43,54);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(28,39)(33,46)(36,50)(40,52)(45,55)(47,53)(49,56)(51,54)

Polycyclic presentation:

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

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, 6405468);