#6498286

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

1254 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)(25,26,30,33)(27,35,36,50)(28,32,40,48)(29,42,43,34)(31,44,45,38)(37,51,39,54)(41,56,52,49)(46,55,47,53);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,29,38,30,36,43,44)(26,31,34,35,33,45,42,50)(28,39,41,55,40,37,52,53)(32,47,49,54,48,46,56,51);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(27,36)(28,40)(31,45)(32,48)(35,50)(38,44)(41,52)(49,56);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,28)(26,32)(27,37)(29,41)(30,40)(31,46)(33,48)(34,49)(35,51)(36,39)(38,53)(42,56)(43,52)(44,55)(45,47)(50,54);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,33)(27,36)(28,40)(34,42)(38,44)(41,52)(46,47)(51,54)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,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, 6405482);