#7019125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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)(3,5,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,28,21,27)(19,29,36,41,31,25,32,38)(20,33,37,42,35,26,34,39)(30,40,46,47,45,43,44,48);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(20,34)(21,23)(25,38)(27,28)(29,41)(30,44)(32,36)(35,37)(39,42)(45,46)(47,48);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,25,28,29)(20,30,35,45)(21,32,23,36)(24,38,27,41)(26,40,33,43)(34,44,37,46)(39,47,42,48);(11,16)(13,15)(17,20,21,34,22,35,23,37)(18,26,27,42,28,33,24,39)(19,30,32,44,31,45,36,46)(25,40,41,48,29,43,38,47);(10,14)(13,15)(19,31)(25,29)(30,45)(32,36)(38,41)(40,43)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=i,d^2=f,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=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6926321);