Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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 | C2×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4440 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)(13,14,18,24)(15,19,27,28)(16,22,17,23)(20,25,21,26);(6,9)(10,12)(13,15)(14,19)(16,26)(17,25)(18,27)(20,22)(21,23)(24,28);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,25)(17,26)(18,27)(19,22)(21,24)(23,28);(13,15)(14,21)(16,25)(17,26)(18,27)(19,23)(20,24)(22,28);(13,16,18,17)(14,22,24,23)(15,25,27,26)(19,21,28,20)Polycyclic presentation:
2^9|a^2=i,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,h^a=hi,e^b=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72250121, #72256996, #72256998, … |
| maximal normal subgroup of | #72250121, #72256996, #72256998, … |
| maximal quotient of | #72305581, #72305589, #72318562 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536428);