#7101080

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1390 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)(19,28)(20,25)(21,35)(22,26)(23,31)(24,30)(27,38)(29,45)(32,48)(33,44)(34,42)(36,40)(37,43)(39,47)(41,46);(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)(25,41,42,45)(29,37,46,33)(39,44,48,43);(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)(32,33)(34,46)(35,40)(36,38)(37,47)(39,44)(42,45)(43,48);(1,3,5,4)(2,7,8,6)(11,16)(13,15)(17,20,22,34)(18,25,26,42)(19,29,30,46)(21,37,38,33)(23,39,40,48)(24,41,28,45)(27,44,35,43)(31,47,36,32);(1,4,5,3)(2,6,8,7)(10,14)(13,15)(18,26)(19,30)(25,42)(27,35)(29,46)(31,36)(32,47)(43,44)

Polycyclic presentation:

2^9|a^2=gi,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=df,e^b=eh,f^b=fi,d^c=dhi,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, 7008276);