Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420056 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15274 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,8,6,12,5,11)(3,13,14,24,17,9,16,21)(4,10,18,25,19,26,20,22)(15,23,29,32,30,27,28,31);(2,8)(3,14)(4,18)(5,7)(9,13)(11,12)(16,17)(19,20)(22,25)(23,31)(27,32)(28,29);(1,3)(2,9)(4,15)(5,16)(6,17)(7,14)(8,21)(10,23)(11,24)(12,13)(18,29)(19,30)(20,28)(22,31)(25,32)(26,27);(1,4,5,18,6,19,7,20)(2,10,11,25,12,26,8,22)(3,15,16,29,17,30,14,28)(9,23,24,32,13,27,21,31)Polycyclic presentation:
2^9|a^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72319466 |
| maximal normal subgroup of | #72319466 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327252);