Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423169 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15312 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60423 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1331 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)(3,8)(4,9)(5,7)(6,11)(10,13)(12,23)(14,28)(15,25)(16,26)(17,24)(18,20)(19,21)(22,27)(29,31)(30,32);(2,7)(3,12)(4,16)(5,13)(8,20)(9,24)(10,11)(14,29)(17,19)(23,25)(27,30)(31,32);(1,3,13,15,6,12,5,18)(2,8,7,20,11,23,10,25)(4,14,16,29,17,27,19,30)(9,22,21,31,24,28,26,32);(1,4)(2,9)(3,14)(5,19)(6,17)(7,21)(8,22)(10,26)(11,24)(12,27)(13,16)(15,29)(18,30)(20,31)(23,28)(25,32)Polycyclic presentation:
2^9|c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72319753 |
| maximal normal subgroup of | #72319753 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330365);