#6448313

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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

1447 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)(19,27)(20,24)(21,25)(22,26)(23,30)(28,39)(29,45)(31,37)(32,48)(33,41)(34,42)(35,43)(36,44)(38,46)(40,47);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,32)(21,30)(22,28)(24,40)(25,27)(26,37)(29,34)(31,35)(33,47)(36,46)(38,42)(39,43)(41,48)(44,45);(9,12,13,11)(10,14,15,16)(19,28)(22,35)(23,37)(26,43)(27,39)(29,32)(30,31)(34,36)(38,40)(42,44)(45,48)(46,47);(17,20,21,33)(18,24,25,41)(19,29,30,46)(22,36,35,34)(23,38,27,45)(26,44,43,42)(28,32,31,47)(37,40,39,48);(1,3,4,7)(2,6,5,8)(18,25)(23,27)(24,41)(26,43)(37,39)(38,45)(40,48)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355509);