Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4322 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)(5,6,9,8)(7,11,12,10)(14,16)(18,19)(22,26)(23,28);(5,7)(6,10)(8,11)(9,12)(13,14,17,16)(15,19,20,18)(21,22)(23,24)(25,26)(27,28);(1,2)(3,4)(13,15)(14,18)(16,19)(17,20)(21,23)(22,27)(24,26)(25,28);(5,8,9,6)(7,11,12,10)(13,16,17,14)(15,19,20,18)(21,24,25,27)(22,23,26,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,23)(22,24)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=fh,b^2=i,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232455, #72232464, #72232560, … |
| maximal normal subgroup of | #72232455, #72232464, #72232560, … |
| maximal quotient of | #72328634, #72330398, #72330865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1176214);