#7160605

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1373 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)(17,18,22,27,23,28,36,39)(19,29,31,26,32,43,21,24)(20,25,35,41,34,40,46,47)(30,44,33,42,45,48,37,38);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,23,32)(18,24,28,26)(20,30,34,45)(21,36,31,22)(25,38,40,42)(27,43,39,29)(33,35,37,46)(41,48,47,44);(2,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,34)(18,25,28,40)(19,30,32,45)(21,37,31,33)(22,35,36,46)(24,38,26,42)(27,41,39,47)(29,44,43,48);(3,8)(5,7)(10,13)(14,16)(17,21,22,19,23,31,36,32)(18,26,27,43,28,24,39,29)(20,33,35,45,34,37,46,30)(25,38,41,44,40,42,47,48);(9,11,12,15)(10,14,13,16)(20,34)(25,40)(30,45)(33,37)(35,46)(38,42)(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,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7067801);