#7093285

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

1380 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,31)(20,34)(21,27)(22,29)(23,28)(24,30)(25,33)(26,37)(32,45)(35,43)(36,44)(38,41)(39,46)(40,47)(42,48);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,38,24,21,23,33)(18,25,28,27,30,41,29,31)(20,35,39,48,37,32,36,47)(26,42,46,43,34,40,44,45);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,24,37)(18,26,30,34)(19,32,21,35)(22,39,23,36)(25,40,41,42)(27,45,31,43)(28,46,29,44)(33,48,38,47);(11,16)(13,15)(17,21,24,19)(18,27,30,31)(20,32,37,35)(22,38,23,33)(25,29,41,28)(26,43,34,45)(36,47,39,48)(40,46,42,44);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,21,38)(20,36,37,39)(25,31,41,27)(26,44,34,46)(32,47,35,48)(40,45,42,43)

Polycyclic presentation:

2^9|b^2=fg,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, 7000481);