Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417566 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(7,16)(9,10)(12,17)(14,24)(18,33)(19,20)(21,22)(25,34)(26,29)(27,30)(31,35)(32,36);(1,2)(3,4)(5,6)(7,17)(8,13)(9,14)(10,24)(11,15)(12,20)(16,26)(18,34)(19,29)(21,27)(22,30)(23,28)(25,31)(32,35)(33,36);(5,7)(6,12)(8,18)(9,20)(10,16)(11,19)(13,25)(14,29)(15,26)(17,24)(21,31)(22,33)(23,35)(27,32)(28,36)(30,34);(2,4)(5,8,11,23)(6,13,15,28)(7,18,19,35)(9,22,10,21)(12,25,26,36)(14,30,24,27)(16,31,20,33)(17,32,29,34)Polycyclic presentation:
2^9|d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72284174, #72284254, #72318018, … |
| maximal normal subgroup of | #72284174, #72284254, #72318018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324763);