Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574008 |
| 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 |
| prime radical | #36361 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,22)(16,21)(17,19)(18,26)(20,25)(23,28)(24,27);(2,4)(5,6,8,9)(7,10,11,12)(14,19)(20,27)(21,26)(22,28);(6,9)(7,11)(13,15,16,24,17,23,18,25)(14,20,26,28,19,27,21,22);(5,7)(6,10)(8,11)(9,12)(15,23)(20,27)(22,28)(24,25);(6,9)(7,11)(13,16,17,18)(14,21,19,26)(15,24,23,25)(20,22,27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307459, #72307516, #72307879, … |
| maximal normal subgroup of | #72307459, #72307516, #72307879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513612);