Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(14,15)(17,18)(20,23)(21,22)(25,26)(27,28);(1,2,3,4)(7,8)(10,12)(13,14,18,21,19,22,17,15)(16,20,26,28,24,27,25,23);(5,7,9,8)(6,10,11,12)(13,15,17,22,19,21,18,14)(16,23,25,27,24,28,26,20);(5,8,9,7)(6,10,11,12)(13,16,19,24)(14,20,22,27)(15,23,21,28)(17,25,18,26);(6,11)(10,12)(16,24)(20,27)(23,28)(25,26)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308085, #72308256, #72308293, … |
| maximal normal subgroup of | #72308085, #72308256, #72308293, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390666);