Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#546 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,23,25,28)(16,21,17,20)(19,26,27,24);(2,4)(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,28)(21,23)(22,27);(6,10)(9,12)(15,24)(16,17)(19,23)(20,21)(25,26)(27,28);(15,24)(16,17)(19,23)(20,21)(25,26)(27,28);(15,25)(19,27)(23,28)(24,26)Polycyclic presentation:
2^9|a^2=i,b^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72263983, #72264025, #72282886, … |
| maximal normal subgroup of | #72263983, #72264025, #72282886, … |
| maximal quotient of | #72264260, #72316300, #72318293 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347614);