#6493916

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=f,e^2=h,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,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, 6401112);