Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,11,15,10,16)(7,17,19,29,21,12,20,28)(8,22,26,32,24,31,25,13)(18,30,35,27,34,36,23,33);(1,2)(3,4)(5,7,11,21)(6,12,15,17)(8,23,24,35)(9,20,10,19)(13,30,32,36)(14,29,16,28)(18,26,34,25)(22,33,31,27);(2,4)(5,8)(6,13)(7,18)(9,26)(10,25)(11,24)(12,27)(14,22)(15,32)(16,31)(17,33)(19,35)(20,23)(21,34)(28,36)(29,30);(6,14)(9,10)(12,28)(13,22)(15,16)(17,29)(19,20)(23,35)(25,26)(27,36)(30,33)(31,32)Polycyclic presentation:
2^9|a^2=ef,b^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286324, #72287003, #72287020, … |
| maximal normal subgroup of | #72286324, #72287003, #72287020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326061);