Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395432 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14724 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,5,13,6,12)(3,8,20,24,16,26,17,25)(4,10,18,9,19,29,21,28)(14,30,15,22,31,27,32,23);(1,3)(2,8)(4,15)(5,16)(6,20)(7,17)(9,27)(10,23)(11,25)(12,24)(13,26)(14,18)(19,32)(21,31)(22,29)(28,30);(1,4)(2,9)(3,14)(5,19)(6,21)(7,18)(8,22)(10,12)(11,29)(13,28)(15,20)(16,31)(17,32)(23,26)(24,27)(25,30);(1,4)(2,10)(3,15)(5,19)(6,18)(7,21)(8,23)(9,12)(11,28)(13,29)(14,20)(16,32)(17,31)(22,26)(24,27)(25,30)Polycyclic presentation:
2^9|a^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215906, #72219682, #72219692, … |
| maximal normal subgroup of | #72215906, #72219682, #72219692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302628);