Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | #32913 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,20)(16,18)(17,19)(21,24)(22,25)(23,26)(27,28);(1,2,3,4)(7,8)(10,11)(14,18)(15,22)(19,21)(23,28);(5,7,9,8)(6,10,12,11)(14,19)(17,24)(18,21)(20,26)(23,28)(25,27);(5,8,9,7)(6,11,12,10)(13,15)(14,20)(16,22)(17,23)(18,25)(19,26)(21,27)(24,28);(6,12)(10,11)(15,22)(20,25)(23,28)(26,27)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394939);