#7162155

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#30549	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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)(21,33)(22,25)(23,29)(26,34)(28,39)(30,40)(31,38)(32,45)(35,41)(36,43)(37,44)(42,47)(46,48);(2,5)(3,7)(10,13)(11,15)(17,19,21,29,22,30,34,43)(18,23,26,27,25,36,33,40)(20,28,32,41,31,42,44,48)(24,35,37,39,38,46,45,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,31)(18,24,25,38)(19,28,30,42)(21,32,34,44)(23,35,36,46)(26,37,33,45)(27,39,40,47)(29,41,43,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,29)(21,34)(23,27)(26,33)(28,41)(30,43)(32,44)(35,39)(36,40)(37,45)(42,48)(46,47);(9,11,12,15)(10,14,13,16)(18,25)(20,31)(23,36)(26,33)(27,40)(28,42)(32,44)(41,48)

Polycyclic presentation:

2^9|b^2=fg,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=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7069351);