Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,28,21,26,22,27)(19,20,30,31,29,24,25,32);(1,3,5,4)(2,6,8,7)(10,13)(14,15)(17,19)(18,24)(20,26)(21,29)(22,25)(23,30)(27,31)(28,32);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(17,20,21,24)(18,25,26,30)(19,27,29,28)(22,32,23,31);(1,4,5,3)(2,7,8,6)(10,13)(14,15)(18,26)(20,24)(27,28)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268754);