Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1428 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)(7,12,18,27)(8,13)(9,16)(10,14)(11,15)(17,33,34,25)(19,26,24,28)(20,29,35,36)(21,32)(22,30)(23,31);(1,2)(3,4)(5,7)(6,12,15,27)(8,20)(9,24)(10,19)(11,18)(13,29,31,36)(14,26,16,28)(17,21)(22,34)(23,35)(25,32,33,30);(2,4)(5,8,11,23)(6,13,15,31)(7,17,18,34)(9,22,10,21)(12,25,27,33)(14,32,16,30)(19,35,24,20)(26,29,28,36);(6,14)(9,10)(12,26)(13,30)(15,16)(17,34)(19,24)(21,22)(25,29)(27,28)(31,32)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^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,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285470, #72285534, #72285600 |
| maximal normal subgroup of | #72285470, #72285534, #72285600 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363856);