#6493122

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1506 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,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, 6400318);