#6565140

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

1353 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,32)(28,40)(29,34)(30,43)(31,35)(33,41)(36,50)(37,46)(38,47)(39,53)(42,48)(44,49)(45,52)(51,56)(54,55);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,31,37)(26,32,35,46)(28,36,42,51)(29,38,30,39)(33,45,49,55)(34,47,43,53)(40,50,48,56)(41,52,44,54);(11,17)(14,20)(18,23)(21,24)(25,28,31,42)(26,33,35,49)(27,36,37,51)(29,41,30,44)(32,45,46,55)(34,48,43,40)(38,52,39,54)(47,56,53,50);(1,3,5,4)(2,7,8,6)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,37)(32,46)(36,51)(38,39)(45,55)(47,53)(50,56)(52,54);(1,4,5,3)(2,6,8,7)(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,43)(27,38,37,39)(28,41,42,44)(32,47,46,53)(33,48,49,40)(36,52,51,54)(45,56,55,50)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=f,e^2=fi,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, 6472336);