#6508215

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1749 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)(17,18,22,27)(19,28,23,24)(20,31,34,37)(21,26,36,42)(25,29,40,32)(30,41,35,38)(33,44,46,47)(39,43,48,45);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,35,36,30)(25,31,40,37)(26,41,42,38)(33,43,46,45)(39,47,48,44);(9,11)(10,13)(12,16)(14,15)(17,20,23,32,22,34,19,29)(18,25,28,31,27,40,24,37)(21,33,35,45,36,46,30,43)(26,39,41,44,42,48,38,47);(11,16)(13,15)(17,21,22,36)(18,26,27,42)(19,30,23,35)(20,33,34,46)(24,38,28,41)(25,39,40,48)(29,43,32,45)(31,44,37,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(21,36)(26,42)(30,35)(33,46)(38,41)(39,48)(43,45)(44,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^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^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415411);