Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2183 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)(8,10)(9,12)(14,19)(15,23)(22,26)(24,25);(5,7,8,12)(6,9,10,11)(13,14,17,19)(15,21,23,28)(16,22,18,26)(20,25,27,24);(1,2,3,4)(13,15)(14,20)(16,24)(17,23)(18,25)(19,27)(21,26)(22,28);(6,10)(9,11)(13,15)(14,21)(16,25)(17,23)(18,24)(19,28)(20,26)(22,27);(7,12)(9,11)(13,16)(14,19)(15,24)(17,18)(20,27)(23,25)Polycyclic presentation:
2^9|b^2=f,c^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256847, #72256948, #72257043, … |
| maximal normal subgroup of | #72256847, #72256948, #72257043, … |
| maximal quotient of | #72264394, #72264422, #72264670 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590014);