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 | #60644 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1968 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)(7,17)(8,14)(9,16)(10,13)(11,15)(12,20)(18,27)(19,33)(21,32)(22,36)(23,31)(24,28)(25,30)(26,34)(29,35);(5,7,9,20,10,21,11,18)(6,12,15,17,13,27,16,32)(8,22,23,35,24,34,25,19)(14,29,30,36,28,33,31,26);(6,13)(7,18,21,20)(12,17,27,32)(14,28)(15,16)(19,22,35,34)(26,33,36,29)(30,31);(2,4)(5,8)(6,14)(7,19)(9,25)(10,24)(11,23)(12,26)(13,28)(15,31)(16,30)(17,33)(18,22)(20,34)(21,35)(27,36)(29,32)Polycyclic presentation:
2^9|b^2=e,c^2=fi,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 | #72285670, #72285809, #72286018, … |
| maximal normal subgroup of | #72285670, #72285809, #72286018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329969);