#6457466

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

1665 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)(19,29)(20,32)(21,25)(22,27)(23,28)(24,38)(26,34)(30,45)(31,41)(33,48)(35,42)(36,40)(37,43)(39,46)(44,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,21)(18,25,28,29)(20,30,35,33)(22,36,24,31)(26,39,43,44)(27,41,38,40)(32,48,42,45)(34,47,37,46);(11,15)(13,16)(17,20,24,34,23,35,22,37)(18,26,27,32,28,43,38,42)(19,30,31,47,21,33,36,46)(25,39,41,48,29,44,40,45);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21,23,19)(18,25,28,29)(20,33,35,30)(22,31,24,36)(26,39,43,44)(27,41,38,40)(32,48,42,45)(34,46,37,47);(17,22,23,24)(18,27,28,38)(19,31,21,36)(20,34,35,37)(25,40,29,41)(26,42,43,32)(30,46,33,47)(39,48,44,45)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,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, 6364662);