#7188531

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1251 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)(4,8)(5,7)(10,14)(12,15)(17,18)(19,27)(20,24)(21,34)(22,25)(23,30)(26,35)(28,41)(29,37)(31,40)(32,38)(33,46)(36,44)(39,45)(42,47)(43,48);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,12,16,15)(17,19)(18,23)(20,28)(21,29)(22,31)(24,36)(25,37)(26,40)(27,34)(30,35)(32,43)(33,42)(38,47)(39,48)(41,46)(44,45);(2,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,32)(18,24,25,38)(19,28,31,43)(21,33,35,45)(23,36,37,47)(26,39,34,46)(27,41,40,48)(29,42,30,44);(1,4,5,3)(2,8,7,6)(10,14)(11,16)(18,25)(19,29)(21,35)(23,27)(24,38)(28,42)(30,31)(33,45)(36,41)(37,40)(43,44)(47,48);(9,12,13,15)(10,11,14,16)(18,25)(20,32)(23,37)(26,34)(27,40)(28,43)(33,45)(42,44)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7095727);