#6497867

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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

1213 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,36)(28,39)(29,32)(30,34)(31,45)(33,42)(35,44)(37,51)(38,47)(40,53)(41,54)(43,50)(46,55)(48,56)(49,52);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,30,29)(26,32,34,36)(28,37,41,40)(31,38,44,43)(33,46,48,49)(35,47,45,50)(39,53,54,51)(42,52,56,55);(11,17)(14,21)(18,23)(22,24)(25,28,31,42,30,41,44,56)(26,33,35,39,34,48,45,54)(27,37,38,52,29,40,43,55)(32,46,47,53,36,49,50,51);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(25,29,30,27)(26,32,34,36)(28,40,41,37)(31,43,44,38)(33,46,48,49)(35,47,45,50)(39,53,54,51)(42,55,56,52);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,29)(32,36)(37,40)(38,43)(46,49)(47,50)(51,53)(52,55)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6405063);