#6457137

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1787 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)(9,10,12,14)(11,13,15,16)(17,18)(19,30)(20,24)(21,26)(22,27)(23,29)(25,34)(28,31)(32,45)(33,40)(35,39)(36,42)(37,44)(38,43)(41,46)(47,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,31,33,23,34,22,20)(18,24,27,25,29,40,28,30)(21,32,36,47,38,46,37,35)(26,39,44,41,43,48,42,45);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,34,23,33,31,19)(18,25,28,24,29,30,27,40)(21,35,37,46,38,47,36,32)(26,41,42,39,43,45,44,48);(11,15)(13,16)(17,21)(18,26)(19,32)(20,35)(22,37)(23,38)(24,39)(25,41)(27,44)(28,42)(29,43)(30,45)(31,36)(33,47)(34,46)(40,48);(17,22,23,31)(18,27,29,28)(19,33,34,20)(21,36,38,37)(24,30,40,25)(26,42,43,44)(32,35,46,47)(39,41,48,45)

Polycyclic presentation:

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

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, 6364333);