#7184815

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

1396 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 7092011);