#6568140

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6475336);