#7125181

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
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	#61185	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=hi,d^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=ehi,d^c=dh,g^c=gi,f^d=fi

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, 7032377);