Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | |
2180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,11,12)(13,14,18,20)(15,23,26,21)(16,19,27,25)(17,28,24,22);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,24,27,18,26,17,16)(14,19,28,21,20,25,22,23);(6,10)(7,11)(13,14)(15,25)(16,21)(17,22)(18,20)(19,26)(23,27)(24,28);(2,4)(6,10)(9,12)(14,20)(15,26)(16,27)(22,28);(13,16,24,15,18,27,17,26)(14,21,28,25,20,23,22,19)Polycyclic presentation:
2^9|a^2=g,b^2=fi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250923, #72257111, #72257116, … |
| maximal normal subgroup of | #72250923, #72257111, #72257116, … |
| maximal quotient of | #72264720 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434749);