Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(5,6)(7,8)(10,14)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,32)(29,31);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,27)(21,26)(23,25)(24,32)(28,29)(30,31);(2,4)(9,11,13,12)(10,15,14,16)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(6,8)(9,12,13,11)(10,15,14,16)(19,27)(22,25)(26,28)(30,32)Polycyclic presentation:
2^9|a^2=fi,c^2=ei,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264950, #72265013, #72265067, … |
| maximal normal subgroup of | #72264950, #72265013, #72265067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327497);