Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455828 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| 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 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,12,15,11,7)(8,17,20,27,22,28,21,13)(9,18,23,14,25,31,24,30)(19,26,34,32,36,29,35,33);(1,2)(3,4)(5,7,10,15,12,16,11,6)(8,13,20,28,22,27,21,17)(9,18,23,30,25,31,24,14)(19,26,34,33,36,29,35,32);(2,4)(5,8,12,22)(6,13,15,27)(7,17,16,28)(9,19,25,36)(10,21,11,20)(14,29,30,26)(18,33,31,32)(23,35,24,34);(5,9)(6,14)(7,18)(8,19)(10,23)(11,24)(12,25)(13,26)(15,30)(16,31)(17,32)(20,34)(21,35)(22,36)(27,29)(28,33)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,f^2=i,g^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,e^b=ei,f^b=fi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285452, #72285534, #72285613 |
| maximal normal subgroup of | #72285452, #72285534, #72285613 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363861);