#8889444

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,23,28)(19,26,21,24)(20,30,29,25)(22,27,31,32)(33,34)(35,41)(36,43)(37,45)(38,40)(39,47)(42,44)(46,48);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,19,23,21)(18,24,28,26)(20,31,29,22)(25,32,30,27)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(2,6)(7,9)(8,12)(14,15)(17,20)(18,25)(19,22)(21,31)(23,29)(24,27)(26,32)(28,30)(33,34,38,40)(35,36,43,41)(37,39,47,45)(42,44,46,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,21,23,19)(18,26,28,24)(20,31,29,22)(25,32,30,27)(33,36,38,41)(34,35,40,43)(37,44,47,48)(39,42,45,46);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,22,23,31)(18,27,28,32)(19,29,21,20)(24,30,26,25)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,48)(46,47);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,30)(31,32)(33,37,38,47)(34,39,40,45)(35,42,43,46)(36,44,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796640);