Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,15,11,17,9,16)(7,18,21,14,22,28,20,12)(8,23,25,31,26,30,24,13)(19,27,34,36,35,29,33,32);(5,7,9,20,11,22,10,21)(6,12,15,18,17,14,16,28)(8,19,24,33,26,35,25,34)(13,29,23,32,31,27,30,36);(2,4)(5,8,11,26)(6,13,17,31)(7,19,22,35)(9,25,10,24)(12,27,14,29)(15,30,16,23)(18,32,28,36)(20,34,21,33);(5,7)(6,14)(8,19)(9,21)(10,20)(11,22)(12,17)(13,29)(15,18)(16,28)(23,36)(24,34)(25,33)(26,35)(27,31)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286567, #72286995, #72287020 |
| maximal normal subgroup of | #72286567, #72286995, #72287020 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326214);