#6611266

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 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:

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

Polycyclic presentation:

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

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, 6518462);