#6480311

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 6387507);