Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15312 |
| prime radical | #60415 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ 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)(3,11)(4,17)(5,21)(6,10)(7,14)(8,19)(9,23)(12,29)(13,16)(15,25)(18,24)(20,27)(22,26)(28,32)(30,31);(1,3)(2,7)(4,18)(5,15)(6,16)(8,24)(9,13)(10,25)(11,21)(12,20)(14,23)(17,28)(19,31)(22,32)(26,29)(27,30);(1,4,6,20)(2,8,10,26)(3,12)(5,22,23,19)(7,24)(9,27,21,17)(11,28)(13,30)(14,32)(15,31)(16,18)(25,29);(1,2)(3,13)(4,8)(5,21)(6,10)(7,15)(9,23)(11,16)(12,30)(14,25)(17,22)(18,28)(19,27)(20,26)(24,31)(29,32)Polycyclic presentation:
2^9|c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72279455, #72284943, #72287280, … |
| maximal normal subgroup of | #72279455, #72284943, #72287280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330380);