#6573289

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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

1356 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)(9,10,15,14)(11,12,13,16)(17,18,22,27,23,28,21,26)(19,29,31,39,32,42,37,24)(20,33,36,40,34,25,35,41)(30,44,46,38,45,43,48,47);(9,11,15,13)(10,16,14,12)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,38)(26,29)(27,42)(28,39)(33,47)(34,45)(35,46)(36,48)(40,43)(41,44);(2,6)(7,8)(9,12,15,16)(10,11,14,13)(17,20,23,34)(18,25,28,33)(19,30,32,45)(21,35,22,36)(24,38,39,47)(26,40,27,41)(29,43,42,44)(31,46,37,48);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,16,14,12)(20,34)(25,33)(30,45)(35,36)(38,47)(40,41)(43,44)(46,48);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,12,13,16)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,35,34,36)(24,29,39,42)(25,40,33,41)(30,46,45,48)(38,43,47,44)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480485);