#6496558

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6403754);