Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420453 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1912 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)(5,6,11,16)(7,17,21,26)(8,13,24,31)(9,15,10,14)(12,18,28,20)(19,33,34,29)(22,36,35,27)(23,32,25,30);(5,7,9,20,11,21,10,18)(6,12,14,17,16,28,15,26)(8,22,23,34,24,35,25,19)(13,29,30,36,31,33,32,27);(7,18,21,20)(12,26,28,17)(19,22,34,35)(27,29,36,33);(2,4)(5,8)(6,13)(7,19)(9,25)(10,23)(11,24)(12,27)(14,32)(15,30)(16,31)(17,33)(18,22)(20,35)(21,34)(26,29)(28,36)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285723, #72285809, #72286090 |
| maximal normal subgroup of | #72285723, #72285809, #72286090 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329970);