#6571108

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478304);