Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#3492 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3138 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)(13,14,18,26,17,22,16,20)(15,21,25,19,24,28,23,27);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,25)(17,24)(18,23)(20,28)(21,26)(22,27);(2,4)(6,9)(7,11)(13,16,17,18)(14,20,22,26)(15,23,24,25)(19,21,27,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,21)(16,23)(17,24)(18,25)(19,26)(20,27)(22,28);(6,9)(7,11)(14,22)(15,24)(20,26)(23,25)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72264047, #72264050, #72264061 |
| maximal normal subgroup of | #72264047, #72264050, #72264061 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503604);