#7166245

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	#60929	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1595 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)(19,31)(20,36)(21,27)(22,30)(23,29)(24,28)(25,34)(26,37)(32,41)(33,46)(35,45)(38,48)(39,43)(40,42)(44,47);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,32)(21,38)(22,40)(23,34)(24,35)(26,41)(27,44)(28,42)(29,45)(30,31)(33,39)(36,46)(37,47)(43,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20,24,21)(18,26,28,43)(19,32,35,38)(22,37,23,39)(25,41,42,48)(27,29,36,30)(31,44,45,46)(33,40,47,34);(3,9)(6,13)(10,15)(14,16)(17,21,24,20)(18,27,28,36)(19,33,35,47)(22,37,23,39)(25,41,42,48)(26,29,43,30)(31,46,45,44)(32,40,38,34);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,28)(25,42)(26,43)(27,36)(29,30)(31,45)(41,48)(44,46)

Polycyclic presentation:

2^9|a^2=h,c^2=i,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, 7073441);