#7101320

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1415 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=g,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dhi,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, 7008516);