Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(14,17)(15,19)(23,27)(25,28);(1,2,3,4)(5,7,9,12)(6,10,11,8)(15,19)(18,20)(21,22,24,26)(23,25,27,28);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,25)(24,27)(26,28);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15,16,19)(14,18,17,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^9|b^2=fghi,c^2=f,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280722 |
| maximal normal subgroup of | #72280722 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3874703);