#6501923

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 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)(3,7)(4,8)(5,6)(17,18,22,27)(19,29,32,24)(20,34,21,26)(23,38,37,28)(25,36,43,35)(30,40,31,44)(33,41,48,45)(39,46,42,47);(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,24,27,29)(20,31,21,30)(23,33,37,48)(25,42,43,39)(26,44,34,40)(28,41,38,45)(35,47,36,46);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,30)(21,22)(23,36)(24,39)(26,28)(27,43)(29,42)(31,32)(33,47)(34,38)(35,37)(40,41)(44,45)(46,48);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(17,21,22,20)(18,26,27,34)(19,31,32,30)(23,36,37,35)(24,40,29,44)(25,38,43,28)(33,47,48,46)(39,45,42,41);(10,14)(13,15)(20,21)(25,43)(26,34)(30,31)(35,36)(39,42)(40,44)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,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, 6409119);