Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2297 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)(5,6,9,15,11,16,10,13)(7,17,20,12,18,28,21,29)(8,22,24,14,25,30,26,32)(19,27,23,33,34,36,35,31);(5,7,9,20,11,18,10,21)(6,12,13,17,16,29,15,28)(8,23,24,34,25,35,26,19)(14,31,22,36,32,33,30,27);(2,4)(6,13)(7,18)(9,10)(12,17)(14,30)(15,16)(19,34)(22,32)(24,26)(27,33)(28,29)(31,36);(5,8,11,25)(6,14,16,32)(7,19,18,34)(9,26,10,24)(12,27,29,36)(13,30,15,22)(17,33,28,31)(20,35,21,23)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=i,e^2=i,f^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,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.
| maximal subgroup of | #72285161, #72285466, #72285534, … |
| maximal normal subgroup of | #72285161, #72285466, #72285534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326213);