Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2846 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)(7,12)(8,10)(9,11)(13,14,18,19)(15,22,17,20)(16,21,25,27)(23,24,26,28);(5,7,9,8)(6,10,11,12)(14,19)(15,20)(17,22)(21,27)(23,28)(24,26);(2,4)(13,15)(14,20)(16,24)(17,18)(19,22)(21,23)(25,28)(26,27);(5,8,9,7)(6,10,11,12)(13,16,18,25)(14,21,19,27)(15,23,17,26)(20,28,22,24);(13,17,18,15)(14,22,19,20)(16,24,25,28)(21,23,27,26)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308549, #72308843 |
| maximal normal subgroup of | #72308549, #72308843 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483729);