#6485928

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1344 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,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, 6393124);