Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456501 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1525 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)(8,13)(9,16)(10,14)(11,15)(12,19)(18,32)(20,31)(21,29)(22,30)(23,28)(24,26)(25,33)(27,34)(35,36);(5,7)(6,12,15,26)(8,18)(9,19)(10,24)(11,20)(13,27,28,25)(14,17,16,31)(21,33)(22,34)(23,35)(29,32,30,36);(2,4)(5,8,11,23)(6,13,15,28)(7,18,20,35)(9,21,10,22)(12,25,26,27)(14,30,16,29)(17,32,31,36)(19,33,24,34);(6,14)(8,21)(9,10)(12,17)(13,28)(15,16)(18,33)(19,24)(22,23)(25,27)(26,31)(34,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,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,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72256458, #72256515, #72256520 |
| maximal normal subgroup of | #72256458, #72256515, #72256520 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371848);