Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| 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 | |
1534 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)(7,16)(8,22)(9,24)(10,27)(11,15)(12,20)(13,26)(14,17)(18,35)(19,23)(21,33)(25,32)(28,29)(30,34)(31,36);(5,7,17,34,11,21,10,20)(6,12,27,33,15,30,14,16)(8,23,28,32,24,18,13,31)(9,25,29,19,22,36,26,35);(5,8,10,28,11,24,17,13)(6,9,14,29,15,22,27,26)(7,18,20,31,21,23,34,32)(12,25,16,19,30,36,33,35);(2,4)(5,9)(6,13)(7,19)(8,14)(10,29)(11,22)(12,23)(15,28)(16,32)(17,26)(18,30)(20,36)(21,35)(24,27)(25,34)(31,33)Polycyclic presentation:
2^9|b^2=ei,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330468);