#6355546

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355483
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1588 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262742);