Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427241 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2471 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15952 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1078 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)(3,11,14,7)(4,15)(5,10)(6,9)(8,17)(12,21,29,27)(13,28,31,22)(16,23,30,32)(18,26)(19,24)(20,25);(1,3)(2,7)(4,16,17,30)(5,13)(6,14)(8,23,15,32)(9,11)(10,22)(12,18,29,19)(20,31)(21,24,27,26)(25,28);(1,4)(2,8,9,15)(3,12)(5,19)(6,17)(7,21,11,27)(10,26,25,24)(13,30)(14,29)(16,31)(18,20)(22,32,28,23);(7,11)(21,27)(22,28)(23,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^b=hi,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237068, #72237078, #72237661, … |
| maximal normal subgroup of | #72237068, #72237078, #72237661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357077);