#7162131

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

Polycyclic presentation:

2^9|b^2=fgi,c^2=i,d^2=g,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, 7069327);