#6507887

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1520 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)(22,36)(23,27)(25,31)(28,40)(29,37)(32,44)(33,38)(34,41)(35,42)(39,45)(43,48)(46,47);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,36)(20,31,33,28)(21,34,35,29)(25,38,40,30)(26,41,42,37)(32,43,46,45)(39,44,48,47);(9,11)(10,13)(12,16)(14,15)(17,20,19,28,23,33,22,31)(18,25,36,38,27,40,24,30)(21,32,29,43,35,46,34,45)(26,39,41,47,42,48,37,44);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,23,35)(18,26,27,42)(19,29,22,34)(20,32,33,46)(24,37,36,41)(25,39,40,48)(28,43,31,45)(30,44,38,47);(10,14)(13,15)(20,33)(21,35)(25,40)(26,42)(28,31)(29,34)(30,38)(37,41)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415083);