#7160589

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1403 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fg,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7067785);