#9040473

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

4137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^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, 8947669);