Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455397 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2700 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16387 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11)(3,13)(4,18)(5,12,6,10)(8,17)(9,21)(14,30,31,26)(15,24)(16,25)(19,23,32,29)(20,27)(22,28);(1,3,5,15,7,17,6,16)(2,8,12,25,11,13,10,24)(4,19,20,31,22,32,21,14)(9,26,18,29,27,30,28,23);(1,4)(2,9,11,27)(3,14)(5,20)(6,21)(7,22)(8,23,13,29)(10,28,12,18)(15,19)(16,32)(17,31)(24,30,25,26);(2,10,11,12)(8,24,13,25)(9,18,27,28)(23,26,29,30)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338619, #72338620, #72338633 |
| maximal normal subgroup of | #72338619, #72338620, #72338633 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368540);