#9023532

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	#54991	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,e^2=g,f^2=gi,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, 8930728);