#9040772

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

4107 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947968);