#6441102

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1723 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,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, 6348298);