#6459209

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1513 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,28,23,29,21,27)(19,30,33,24,34,41,32,42)(20,25,38,45,36,26,37,44)(31,46,35,39,47,43,48,40);(2,6)(7,8)(17,19,23,34)(18,24,29,42)(20,35,36,48)(21,33,22,32)(25,43,26,46)(27,41,28,30)(31,37,47,38)(39,44,40,45);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,38,23,36,21,37)(18,25,28,45,29,26,27,44)(19,31,33,35,34,47,32,48)(24,39,41,43,42,40,30,46);(1,4,5,3)(2,8,6,7)(17,20,21,37,23,36,22,38)(18,26,27,45,29,25,28,44)(19,31,32,48,34,47,33,35)(24,40,30,43,42,39,41,46);(9,11,12,15)(10,14,13,16)(20,36)(25,26)(31,47)(35,48)(37,38)(39,40)(43,46)(44,45)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei,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, 6366405);