Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,11,12)(13,14)(15,23)(16,24)(17,18)(19,21)(20,26)(22,28)(25,27);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,24,19,25,28,20)(14,16,22,23,21,26,17,27);(6,10)(7,11)(13,16,19,26)(14,20,21,24)(15,17,25,22)(18,23,28,27);(2,4)(6,10)(9,12)(14,21)(16,26)(17,22)(23,27);(13,17,28,21,19,22,18,14)(15,23,20,16,25,27,24,26)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=i,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250923, #72257086, #72257092 |
| maximal normal subgroup of | #72250923, #72257086, #72257092 |
| maximal quotient of | #72264671 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434752);