#7101069

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

1401 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)(19,24)(20,25)(21,34)(22,26)(23,30)(27,37)(28,40)(29,41)(31,48)(32,45)(33,43)(35,39)(36,44)(38,47)(42,46);(9,10)(11,15)(12,14)(13,16)(17,19,22,29)(18,23,26,39)(20,31,33,47)(21,35,37,30)(24,27,41,34)(25,42,43,40)(28,36,46,32)(38,45,48,44);(1,3,5,4)(2,6,8,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,28)(21,30)(22,29)(23,34)(25,40)(26,41)(27,39)(31,32)(33,46)(35,37)(36,47)(38,44)(42,43)(45,48);(1,3,5,4)(2,7,8,6)(11,16)(13,15)(17,20,22,33)(18,25,26,43)(19,28,29,46)(21,36,37,32)(23,38,39,48)(24,40,41,42)(27,45,34,44)(30,47,35,31);(1,4,5,3)(2,6,8,7)(10,14)(13,15)(18,26)(19,29)(25,43)(27,34)(28,46)(30,35)(31,47)(44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=gi,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,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, 7008265);