Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456510 |
| 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×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,18)(8,13)(9,24,27,33)(10,16)(11,17)(12,20)(14,26,34,28)(15,22)(19,29,23,32)(21,30)(25,31,36,35);(1,2)(3,4)(5,7,10,20,11,8,22,21)(6,12,15,18,17,30,16,13)(9,19,28,25,27,23,26,36)(14,29,24,35,34,32,33,31);(2,4)(5,8)(6,13)(7,11)(9,25)(10,20)(12,16)(14,32)(15,30)(17,18)(19,28)(21,22)(23,26)(24,35)(27,36)(29,34)(31,33);(5,9)(6,14)(7,19)(8,23)(10,28)(11,27)(12,29)(13,31)(15,24)(16,33)(17,34)(18,35)(20,25)(21,36)(22,26)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=ef,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,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363828);