Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,12,8,10)(13,14,18,19)(15,16,22,24)(17,25,27,20)(21,26,28,23);(5,6)(7,10)(8,12)(9,11)(13,15)(14,16)(17,23)(18,22)(19,24)(20,28)(21,25)(26,27);(1,2,3,4)(13,16,14,15,18,24,19,22)(17,26,20,28,27,23,25,21);(5,7,9,8)(6,10,11,12)(13,14,18,19)(15,16,22,24)(17,20,27,25)(21,23,28,26);(5,8,9,7)(6,10,11,12)(13,17,14,20,18,27,19,25)(15,21,24,26,22,28,16,23)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280952, #72283038, #72283309, … |
| maximal normal subgroup of | #72280952, #72283038, #72283309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724052);