Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,13,14,20)(8,16,18,11)(10,17,19,15)(21,22,26,25)(23,27,31,30)(24,28,33,32)(29,35,36,34);(5,7)(6,10)(8,17)(9,14)(11,20)(12,19)(13,16)(15,18)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36);(21,24)(22,28)(23,29)(25,32)(26,33)(27,34)(30,35)(31,36);(2,4)(7,14)(10,19)(13,20)(15,17)(21,25,26,22)(23,30,31,27)(24,32,33,28)(29,35,36,34)Polycyclic presentation:
2^9|a^2=hi,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,d^b=dh,e^b=eh,f^b=fi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72268065 |
| maximal normal subgroup of | #72268065 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253586);