#6462570

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 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:

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

Polycyclic presentation:

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

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, 6369766);