#8943821

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	#50927	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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)(3,9)(4,7)(5,13)(6,8)(10,14)(11,15)(12,16)(17,18,23,27)(19,26,31,24)(20,25,21,22)(28,30,29,32)(33,34,38,41)(35,39,45,48)(36,40,46,37)(42,43,44,47);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46);(2,8)(5,14)(7,9)(11,12)(17,20,23,21)(18,25,27,22)(19,28,31,29)(24,32,26,30)(33,36,38,46)(34,40,41,37)(35,42,45,44)(39,43,48,47);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,21,23,20)(18,25,27,22)(19,29,31,28)(24,32,26,30)(33,37)(34,36)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,19)(18,26)(20,29)(21,28)(22,30)(23,31)(24,27)(25,32)(33,36)(34,40)(35,44)(37,41)(38,46)(39,47)(42,45)(43,48);(1,5,6,14)(2,10,8,13)(3,11,4,12)(7,15,9,16)(17,22,23,25)(18,20,27,21)(19,30,31,32)(24,28,26,29)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851017);