Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,18)(16,25)(17,21)(19,22)(20,23)(24,27)(26,28);(1,2,3,4)(6,9)(10,11)(13,15,18,14,20,22,19,23)(16,21,27,24,17,25,26,28);(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,24)(17,20)(18,26)(19,27)(22,28)(23,25);(13,17)(14,21)(15,24)(16,20)(18,27)(19,26)(22,28)(23,25);(13,18,20,19)(14,22,23,15)(16,26,17,27)(21,28,25,24)Polycyclic presentation:
2^9|b^2=fg,c^2=gi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72283529 |
| maximal normal subgroup of | #72283529 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353877);