#6482343

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1347 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,36)(28,38)(29,34)(30,35)(31,32)(33,37)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,30,31)(26,32,35,36)(28,39,29,37)(33,34,40,38);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(25,28,30,29)(26,33,35,40)(27,37,31,39)(32,38,36,34);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,30,28)(26,34,35,38)(27,37,31,39)(32,40,36,33);(10,16)(14,21)(15,24)(22,23)(25,27,30,31)(26,32,35,36)(28,37,29,39)(33,38,40,34)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,g^c=gi,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, 6389539);