#6441489

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348685);