#6468090

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1626 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)(9,10,13,16)(11,14,12,15)(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)(43,48)(44,47);(11,12)(14,15)(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,42)(20,29,35,45,33,43,34,44)(25,37,40,41,39,47,46,48);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,41)(30,44)(31,45)(32,43)(36,46)(38,47)(42,48);(9,12,13,11)(10,15,16,14)(20,33)(25,39)(29,43)(34,35)(37,47)(40,46)(41,48)(44,45);(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,33,35)(24,28,38,42)(25,40,39,46)(29,44,43,45)(37,41,47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=gh,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375286);