#6448562

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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1449 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,29)(20,26)(21,27)(22,25)(23,28)(24,32)(30,38)(31,45)(33,40)(34,48)(35,43)(36,41)(37,44)(39,47)(42,46);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,34)(21,33)(22,32)(23,30)(25,29)(26,42)(27,38)(28,40)(31,37)(35,47)(36,46)(39,44)(41,48)(43,45);(9,12,13,11)(10,14,15,16)(18,25)(19,30)(21,23)(24,38)(26,41)(29,40)(31,46)(32,33)(34,47)(35,37)(39,42)(45,48);(17,20,22,36)(18,26,25,41)(19,31,32,47)(21,35,23,37)(24,39,29,45)(27,43,28,44)(30,46,33,34)(38,42,40,48);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,25,28)(19,30,32,33)(20,35,36,37)(24,40,29,38)(26,43,41,44)(31,46,47,34)(39,48,45,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,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, 6355758);