Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806551 |
| 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 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,25,26)(23,27,30,34)(24,31,33,28)(29,36,35,32);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,27)(24,32)(25,30)(26,34)(28,35)(29,31)(33,36);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,26,31,25,33,22,28)(23,29,27,32,30,35,34,36);(6,12)(10,13)(11,18)(17,19)(21,22,25,26)(23,27,30,34)(24,28,33,31)(29,32,35,36);(7,15)(10,13)(14,20)(17,19)(21,24,22,28,25,33,26,31)(23,29,34,36,30,35,27,32)Polycyclic presentation:
2^9|a^2=i,c^2=fh,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257163, #72257166, #72257204, … |
| maximal normal subgroup of | #72257163, #72257166, #72257204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714460);