#7162141

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1513 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)(17,18,22,25)(19,27,32,38)(20,33,34,24)(21,36,28,26)(23,30,39,31)(29,43,45,41)(35,40,42,48)(37,46,47,44);(2,5)(3,7)(10,13)(11,15)(17,19,28,30,22,32,21,31)(18,23,36,38,25,39,26,27)(20,29,42,44,34,45,35,46)(24,37,48,43,33,47,40,41);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,34)(18,24,25,33)(19,29,32,45)(21,35,28,42)(23,37,39,47)(26,40,36,48)(27,41,38,43)(30,44,31,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,30)(21,28)(23,38)(26,36)(27,39)(29,44)(31,32)(35,42)(37,43)(40,48)(41,47)(45,46);(9,11,12,15)(10,14,13,16)(18,25)(20,34)(23,39)(26,36)(27,38)(29,45)(35,42)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=gi,e^2=h,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, 7069337);