#8406852

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	#52451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8314048);