Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12);(2,4)(13,14)(15,21)(16,24)(17,20)(18,23)(19,26)(22,27)(25,28);(6,9)(10,12)(13,15)(14,18)(16,25)(17,22)(19,27)(20,28)(21,24)(23,26);(14,19)(16,26)(18,27)(20,24)(21,28)(23,25);(5,7,8,11)(6,10,9,12)(15,22)(18,28)(21,27)(23,25)Polycyclic presentation:
2^9|a^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72307116, #72307139, #72307412 |
| maximal normal subgroup of | #72307116, #72307139, #72307412 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361669);