#7184876

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=i,e^2=hi,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, 7092072);