Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418976 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,16)(8,21,24,32)(9,26)(10,14)(11,15)(12,19)(13,25,31,23)(17,29)(18,28,34,36)(20,27)(22,33,35,30);(1,2,3,4)(5,7,9,19,11,20,10,17)(6,12,14,16,15,29,26,27)(8,22,23,34,24,35,25,18)(13,30,21,36,31,33,32,28);(7,17)(9,10)(12,27)(14,26)(16,29)(18,22)(19,20)(21,32)(23,25)(28,30)(33,36)(34,35);(2,4)(5,8)(6,13)(7,18)(9,25)(10,23)(11,24)(12,28)(14,32)(15,31)(16,33)(17,22)(19,35)(20,34)(21,26)(27,30)(29,36)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,g^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,e^c=ei,g^c=gi,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, 326172);