#6497018

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 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)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(27,34)(28,29)(31,39)(33,37)(35,40)(36,38);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,26,30,32)(27,35,36,31)(28,37,29,33)(34,40,38,39);(11,17)(14,21)(18,23)(22,24)(25,27,30,36)(26,31,32,35)(28,34,29,38)(33,40,37,39);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(26,32)(31,35)(33,37)(39,40);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,28,30,29)(26,33,32,37)(27,34,36,38)(31,40,35,39)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=i,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,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, 6404214);