Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
3264 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)(13,14,17,20)(15,19,16,18)(21,22)(23,24)(25,27)(26,28);(5,6,8,10)(7,9,12,11)(21,23,25,28)(22,26,27,24);(7,12)(9,11)(13,15,17,16)(14,18,20,19)(21,24,25,26)(22,23,27,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,16,17,15)(14,19,20,18)(21,24)(22,23)(25,26)(27,28);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,24)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=gh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72333832, #72333909, #72333930 |
| maximal normal subgroup of | #72333832, #72333909, #72333930 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 996355);