Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3038 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2714 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)(7,12)(8,10)(9,11)(13,14,18,21)(15,22,20,17)(16,19,24,26)(23,28,27,25);(7,8)(10,12)(13,15,17,14,18,20,22,21)(16,23,25,19,24,27,28,26);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,19)(15,23)(17,25)(18,24)(20,27)(21,26)(22,28);(5,8,9,7)(6,10,11,12);(16,24)(19,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308676, #72308686, #72308722, … |
| maximal normal subgroup of | #72308676, #72308686, #72308722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370286);