Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,12,19,29)(8,21,24,34)(9,14,20,28)(10,16)(11,15)(13,25,32,26)(17,33)(18,31)(22,36)(23,30)(27,35);(1,2,3,4)(5,7,17,20,11,19,10,9)(6,12,16,14,15,29,33,28)(8,22,25,23,24,18,26,35)(13,30,34,36,32,27,21,31);(2,4)(5,8)(6,13)(7,18)(9,23)(10,26)(11,24)(12,27)(14,31)(15,32)(16,34)(17,25)(19,22)(20,35)(21,33)(28,36)(29,30);(5,9,10,19,11,20,17,7)(6,14,33,12,15,28,16,29)(8,23,26,22,24,35,25,18)(13,31,21,27,32,36,34,30)Polycyclic presentation:
2^9|a^2=h,b^2=efg,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269838);