#7162150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,22,26)(19,27,30,39)(20,32,34,24)(21,36,37,25)(23,29,40,31)(28,44,33,43)(35,42,45,48)(38,46,41,47);(2,5)(3,7)(10,13)(11,15)(17,19,21,31,22,30,37,29)(18,23,25,27,26,40,36,39)(20,33,35,47,34,28,45,46)(24,41,42,44,32,38,48,43);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,34)(18,24,26,32)(19,28,30,33)(21,35,37,45)(23,38,40,41)(25,42,36,48)(27,43,39,44)(29,47,31,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,29)(20,34)(21,37)(23,39)(24,32)(25,36)(27,40)(28,46)(30,31)(33,47)(38,43)(41,44);(9,11,12,15)(10,14,13,16)(19,30)(20,34)(23,40)(24,32)(27,39)(29,31)(35,45)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=i,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7069346);