#6495010

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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)(11,15)(14,16)(17,18)(19,27)(20,24)(21,34)(22,25)(23,30)(26,35)(28,41)(29,42)(31,46)(32,39)(33,48)(36,44)(37,45)(38,47)(40,43);(9,10)(11,14)(12,13)(15,16)(17,19,21,30,22,29,35,45)(18,23,26,27,25,37,34,42)(20,31,33,47,32,28,43,44)(24,38,40,46,39,36,48,41);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,39)(26,40)(27,41)(29,31)(30,44)(34,48)(35,43)(37,38)(42,46)(45,47);(10,13)(14,16)(19,29)(20,32)(23,37)(24,39)(27,42)(30,45)(33,43)(40,48);(9,11,12,15)(10,14,13,16)(18,25)(23,37)(24,39)(26,34)(27,42)(36,38)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di

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, 6402206);