Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#1999 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2984 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)(17,18)(19,23)(20,22)(21,25)(24,28)(26,30)(27,29)(31,32);(9,10)(11,14)(12,13)(15,16)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(2,4)(10,13)(14,16)(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,30)(28,32);(6,8)(9,11,12,15)(10,14,13,16)(19,26)(23,30)(27,32)(29,31)Polycyclic presentation:
2^9|d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264208, #72264220, #72264320, … |
| maximal normal subgroup of | #72264208, #72264220, #72264320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311110);