#6435320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1581 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)(11,15)(14,16)(17,18)(19,23)(20,30)(21,34)(22,25)(24,32)(26,36)(27,43)(28,38)(29,46)(31,48)(33,42)(35,41)(37,44)(39,45)(40,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,23,25,38)(20,31,33,27)(21,29,36,45)(24,40,41,37)(26,39,34,46)(30,48,42,43)(32,47,35,44);(10,13)(14,16)(17,20,22,33)(18,24,25,41)(19,27,28,31)(21,35,36,32)(23,37,38,40)(26,42,34,30)(29,47,45,44)(39,48,46,43);(9,11,12,15)(10,14,13,16)(18,25)(23,38)(24,41)(26,34)(30,42)(37,40)(39,46)(43,48);(1,3,4,7)(2,6,5,8)(19,28)(23,38)(27,31)(29,45)(37,40)(39,46)(43,48)(44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342516);