#9017170

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#43383	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=i,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924366);