Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1722 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)(9,10,13,12)(11,15,16,14)(17,18,22,27)(19,28,31,32)(20,26,21,25)(23,30,24,29);(9,11,13,16)(10,14,12,15)(17,19,20,29,22,31,21,30)(18,23,25,28,27,24,26,32);(2,4)(10,12)(14,15)(19,29)(20,21)(23,28)(24,32)(25,26)(30,31);(6,8)(9,12,13,10)(11,15,16,14)(17,19,21,30,22,31,20,29)(18,24,26,28,27,23,25,32)Polycyclic presentation:
2^9|a^2=egi,b^2=e,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264366, #72264400, #72264513, … |
| maximal normal subgroup of | #72264366, #72264400, #72264513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367576);