#9026654

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

4091 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8933850);