Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 |
| prime radical | C2×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1745 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(22,27)(23,31)(30,34)(32,33);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,22)(23,29)(24,30)(25,27)(26,34)(28,32)(31,36)(33,35);(1,2,3,4)(21,23)(22,28)(24,32)(25,31)(26,33)(27,35)(29,34)(30,36);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,23)(22,29)(24,33)(25,31)(26,32)(27,36)(28,34)(30,35);(7,15)(10,13)(14,20)(18,19)(21,24)(22,27)(23,32)(25,26)(28,35)(31,33)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308226 |
| maximal normal subgroup of | #72308226 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591390);