Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | |
2184 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)(14,19)(16,24)(17,26)(20,27)(21,23)(25,28);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,14,17,19,18,20,26,27)(15,16,23,24,22,25,21,28);(6,10)(7,11)(13,15)(14,16)(17,23)(18,22)(19,24)(20,25)(21,26)(27,28);(2,4)(6,10)(9,12)(14,20)(15,22)(19,27)(21,23);(13,16,17,24,18,25,26,28)(14,21,19,15,20,23,27,22)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=f,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^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283139, #72283615, #72285546 |
| maximal normal subgroup of | #72283139, #72283615, #72285546 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434675);