#6485903

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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)(11,15)(12,16)(13,14)(17,18)(19,26)(20,25)(21,27)(22,30)(23,29)(24,28)(31,42)(32,40)(33,41)(34,39)(35,48)(36,46)(37,45)(38,44)(43,47);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,22,20,24,33,23,32)(18,25,29,26,28,40,30,41)(21,31,37,34,36,47,38,35)(27,39,44,42,46,48,45,43);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,22,33,24,32,23,19)(18,26,29,40,28,41,30,25)(21,35,37,31,36,34,38,47)(27,43,44,39,46,42,45,48);(9,11,13,12)(10,15,14,16)(17,21)(18,27)(19,31)(20,34)(22,37)(23,38)(24,36)(25,39)(26,42)(28,46)(29,44)(30,45)(32,35)(33,47)(40,48)(41,43);(1,4,5,3)(2,6,8,7)(9,12,13,11)(10,15,14,16)(18,28)(21,36)(25,40)(26,41)(29,30)(31,47)(34,35)(37,38)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=hi,e^2=fg,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=dhi,d^c=di,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, 6393099);