Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(22,26)(27,35)(28,33)(29,36);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,22,24,26)(23,29,30,36)(25,33,32,28)(27,34,35,31);(1,2,3,4)(21,23,24,30)(22,27,26,35)(25,34,32,31)(28,36,33,29);(6,11)(10,13)(12,19)(18,20)(23,30)(27,35)(29,36)(31,34);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,28)(25,32)(26,33)(27,36)(29,35)(31,34)Polycyclic presentation:
2^9|b^2=f,c^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281541, #72282270, #72282323, … |
| maximal normal subgroup of | #72281541, #72282270, #72282323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593397);