#6507465

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1519 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,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, 6414661);