#6490244

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1342 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:

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

Polycyclic presentation:

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

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, 6397440);