#7091765

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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=ghi,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6998961);