#9160809

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

4168 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,23,25)(19,26,22,24)(20,29,30,21)(27,32,31,28)(33,34)(35,44)(36,41)(37,38)(39,43)(40,47)(42,46)(45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19,23,22)(18,24,25,26)(20,27,30,31)(21,32,29,28)(33,35)(34,40)(36,45)(37,46)(38,41)(39,47)(42,48)(43,44);(2,6)(7,9)(8,12)(15,16)(18,25)(20,30)(24,26)(27,31)(33,36,39,46)(34,41,43,42)(35,45,47,37)(38,44,48,40);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,20,23,30)(18,21,25,29)(19,27,22,31)(24,32,26,28)(33,36)(34,42)(35,45)(37,47)(38,44)(39,46)(40,48)(41,43);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,21)(18,20)(19,28)(22,32)(23,29)(24,31)(25,30)(26,27)(33,37,39,45)(34,38,43,48)(35,46,47,36)(40,41,44,42);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,22,23,19)(18,26,25,24)(20,31,30,27)(21,28,29,32)(33,38)(34,37)(35,41)(36,44)(39,48)(40,46)(42,47)(43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9068005);