Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,11,17,23)(8,12)(9,13,20,27)(10,14)(15,28,30,21)(16,31)(18,24,34,26)(19,25)(22,33)(29,35)(32,36);(1,2)(3,4)(5,7,10,17)(6,11)(8,15)(9,16,20,33)(12,21,25,28)(13,22)(14,23)(18,29)(19,30)(24,35,26,36)(27,31)(32,34);(2,4)(5,8)(6,12)(7,15,17,30)(9,18,20,34)(10,19)(11,21,23,28)(13,26,27,24)(14,25)(16,32)(22,35)(29,33)(31,36);(5,9)(6,13,14,27)(7,16)(8,18,19,34)(10,20)(11,22,23,31)(12,24)(15,29,30,32)(17,33)(21,35)(25,26)(28,36)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=f,d^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72283644, #72283700, #72283846, … |
| maximal normal subgroup of | #72283644, #72283700, #72283846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367293);