#7166241

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60931	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi

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, 7073437);