#7166250

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	#60933	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)(19,29)(20,35)(21,26)(22,39)(23,28)(24,33)(25,38)(27,40)(30,44)(31,47)(32,43)(34,42)(36,41)(37,45)(46,48);(3,9)(4,12)(6,13)(7,11)(17,19)(18,24)(20,36)(21,30)(22,34)(23,32)(25,44)(26,46)(27,43)(28,42)(29,39)(31,38)(33,40)(35,47)(37,48)(41,45);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20)(18,25)(19,30)(21,23)(22,37)(24,41)(26,27)(28,45)(29,46)(31,34)(32,36)(33,48)(35,39)(38,40)(42,44)(43,47);(3,9)(6,13)(10,15)(14,16)(17,21,23,20)(18,26,28,35)(19,31,32,48)(22,37,40,38)(24,41,42,44)(25,39,45,27)(29,47,43,46)(30,33,36,34);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(19,32)(24,42)(29,43)(30,36)(31,48)(33,34)(41,44)(46,47)

Polycyclic presentation:

2^9|a^2=hi,d^2=i,e^2=hi,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, 7073446);