#9160812

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

4157 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,34,39,37)(35,44,46,40)(36,41,47,42)(38,43,48,45);(2,7)(4,15)(8,11)(10,13)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,35,39,46)(34,40,37,44)(36,45,47,43)(38,41,48,42);(2,7)(3,12)(9,14)(10,13)(18,22)(23,25)(24,28)(31,32)(33,36,39,47)(34,41,37,42)(35,45,46,43)(38,40,48,44);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,36)(34,42)(35,45)(37,41)(38,40)(39,47)(43,46)(44,48);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,37,39,34)(35,44,46,40)(36,42,47,41)(38,43,48,45);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,38)(34,43)(35,41)(36,40)(37,45)(39,48)(42,46)(44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,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, 9068008);