#7145494

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	#45388	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1694 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)(4,5)(7,8)(11,15)(14,16)(17,18,24,29)(19,25,34,42)(20,27,21,36)(22,28,23,38)(26,37,43,35)(30,40,31,45)(32,41,33,47)(39,46,48,44);(2,5)(3,7)(10,13)(11,15)(17,19,24,34)(18,25,29,42)(20,30,21,31)(22,32,23,33)(26,39,43,48)(27,40,36,45)(28,41,38,47)(35,44,37,46);(9,10)(11,14)(12,13)(15,16)(17,20,24,21)(18,26,29,43)(19,30,34,31)(22,35,23,37)(25,39,42,48)(27,28,36,38)(32,44,33,46)(40,41,45,47);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,24,20)(18,27,29,36)(19,31,34,30)(22,35,23,37)(25,40,42,45)(26,28,43,38)(32,44,33,46)(39,41,48,47);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,28,29,38)(19,32,34,33)(20,35,21,37)(25,41,42,47)(26,36,43,27)(30,44,31,46)(39,45,48,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7052690);