#6457921

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365117);