Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453994 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16297 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1624 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)(3,11,13,7)(4,8)(5,18)(6,9)(10,20)(12,21,28,27)(14,22,19,26)(15,29,30,23)(16,31)(17,24)(25,32);(1,3)(2,7)(4,15)(5,19)(6,13)(8,23)(9,11)(10,26)(12,16)(14,20)(17,30)(18,22)(21,25)(24,29)(27,31)(28,32);(1,4)(2,8)(3,12,13,28)(5,16)(6,17)(7,21,11,27)(9,24)(10,25)(14,15,19,30)(18,31)(20,32)(22,23,26,29);(8,24)(21,27)(23,29)(25,31)Polycyclic presentation:
2^9|a^2=f,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237064, #72237069, #72237661, … |
| maximal normal subgroup of | #72237064, #72237069, #72237661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364779);