Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417487 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2715 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)(7,17,20,25)(8,13,23,29)(9,15,10,14)(12,18,27,21)(19,31,34,35)(22,30,24,28)(26,32,36,33);(7,18,20,21)(12,25,27,17)(19,33,34,32)(26,31,36,35);(5,7)(6,12)(8,19)(9,18)(10,21)(11,20)(13,26)(14,25)(15,17)(16,27)(22,32)(23,34)(24,33)(28,35)(29,36)(30,31);(2,4)(5,8)(6,13)(7,19)(9,22)(10,24)(11,23)(12,26)(14,28)(15,30)(16,29)(17,31)(18,32)(20,34)(21,33)(25,35)(27,36)Polycyclic presentation:
2^9|a^2=i,b^2=e,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,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286018, #72286090, #72286176 |
| maximal normal subgroup of | #72286018, #72286090, #72286176 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325039);