Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,24)(19,21,20,23)(25,26)(27,30)(28,31)(29,32);(2,3)(4,8)(10,13)(14,16)(17,19)(18,23)(20,22)(21,24)(25,27)(26,29)(28,32)(30,31);(2,3)(6,7)(11,15)(14,16)(18,24)(19,20)(25,27,28,32)(26,30,31,29);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,22,19)(18,21,24,23)(25,27)(26,30)(28,32)(29,31);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,19)(20,24)(22,23)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72326596, #72326602, #72326653, … |
| maximal normal subgroup of | #72326596, #72326602, #72326653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1493629);