Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14)(15,19)(16,18)(17,21)(20,24)(22,27)(23,25)(26,28);(1,2,4,3)(5,6)(7,10)(8,9)(11,12);(1,3,4,2)(6,9)(7,11)(14,18)(15,22)(16,24)(19,26)(20,21)(25,27);(5,7,8,11)(6,10,9,12)(13,15,17,22)(14,19,21,27)(16,23,24,28)(18,25,20,26);(6,9)(10,12)(15,22)(19,27)(23,28)(25,26)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308401, #72308800 |
| maximal normal subgroup of | #72308401, #72308800 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421658);