Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417805 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15258 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13)(5,6)(10,22)(11,12)(14,24)(15,29)(16,17)(18,26)(19,20)(23,32)(27,30);(1,2,5,8,7,12,6,11)(3,14,13,25,17,24,16,9)(4,18,19,26,21,22,20,10)(15,23,29,31,30,32,27,28);(1,3)(2,9)(4,15)(5,13)(6,16)(7,17)(8,14)(10,23)(11,24)(12,25)(18,31)(19,29)(20,27)(21,30)(22,28)(26,32);(1,4)(2,10)(3,15)(5,20)(6,19)(7,21)(8,22)(9,23)(11,18)(12,26)(13,27)(14,28)(16,29)(17,30)(24,31)(25,32)Polycyclic presentation:
2^9|b^2=e,e^2=i,f^2=i,g^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,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72215968, #72215995, … |
| maximal normal subgroup of | #72213514, #72215968, #72215995, … |
| maximal quotient of | #72250596, C2×#417805, #72252145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325001);