#6570242

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1361 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)(4,6)(5,8)(9,10,15,14)(11,12,13,16)(17,18,22,27)(19,28,30,24)(20,32,34,47)(21,26,23,37)(25,33,41,35)(29,43,44,46)(31,40,36,39)(38,45,42,48);(1,3,5,4)(2,6,8,7)(9,11,15,13)(10,16,14,12)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,40)(27,28)(32,46)(33,48)(34,44)(35,45)(37,39)(41,42)(43,47);(9,12,15,16)(10,11,14,13)(17,20,23,35,22,34,21,33)(18,25,26,32,27,41,37,47)(19,29,31,45,30,44,36,48)(24,38,40,46,28,42,39,43);(1,3,5,4)(2,7,8,6)(9,13,15,11)(10,16,14,12)(19,30)(24,28)(29,44)(31,36)(38,42)(39,40)(43,46)(45,48);(1,4,5,3)(2,6,8,7)(9,14,15,10)(11,12,13,16)(17,21,22,23)(18,26,27,37)(19,31,30,36)(20,33,34,35)(24,39,28,40)(25,32,41,47)(29,45,44,48)(38,43,42,46)

Polycyclic presentation:

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

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, 6477438);