#9206539

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

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, 9113735);