#6448215

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1617 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)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,26)(21,27)(22,25)(23,28)(29,39)(30,38)(31,40)(32,37)(33,42)(34,43)(35,41)(36,44)(45,48)(46,47);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,22,31)(18,24,25,40)(20,33,35,46)(21,32,23,29)(26,42,41,47)(27,37,28,39)(30,36,45,34)(38,44,48,43);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,14,15,16)(18,25)(19,29)(21,23)(24,37)(26,41)(30,33)(31,32)(34,36)(38,47)(39,40)(42,48)(45,46);(1,4,5,3)(2,6,8,7)(17,20)(18,26)(19,30)(21,34)(22,35)(23,36)(24,38)(25,41)(27,43)(28,44)(29,33)(31,45)(32,46)(37,47)(39,42)(40,48);(17,21,22,23)(18,27,25,28)(19,29,31,32)(20,34,35,36)(24,39,40,37)(26,43,41,44)(30,33,45,46)(38,42,48,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,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, 6355411);