#6529914

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
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

1595 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:

(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);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(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);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(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);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(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);(2,8)(6,9)(7,16)(14,15)(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|b^2=f,c^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=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, 6437110);