Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453847 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16152 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15,18,8)(5,19)(6,10)(7,14)(9,21)(12,26,16,22)(13,23)(17,24,30,28)(20,25)(27,31,29,32);(1,3)(2,7,10,11)(4,16)(5,20,21,13)(6,14)(8,22,15,26)(9,25)(12,18)(17,27,30,29)(19,23)(24,32)(28,31);(1,4)(2,8)(3,12,14,16)(5,17)(6,18)(7,22,11,26)(9,24)(10,15)(13,27,20,29)(19,28)(21,30)(23,31,25,32);(1,5)(2,9,10,19)(3,13)(4,17)(6,21)(7,23,11,25)(8,24,15,28)(12,27)(14,20)(16,29)(18,30)(22,31,26,32)Polycyclic presentation:
2^9|a^2=efg,b^2=efh,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72317862, #72317878, #72317890, … |
| maximal normal subgroup of | #72317862, #72317878, #72317890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361043);