Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3402 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)(5,6,9,15,10,14,26,13)(7,16,24,29,19,28,18,11)(8,20,22,31,23,12,25,32)(17,30,34,27,33,35,21,36);(2,4)(5,7)(6,11)(8,21)(9,24)(10,19)(12,30)(13,28)(14,29)(15,16)(17,22)(18,26)(20,35)(23,34)(25,33)(27,32)(31,36);(5,8,10,23)(6,12,14,20)(7,17,19,33)(9,25,26,22)(11,27,29,36)(13,32,15,31)(16,30,28,35)(18,34,24,21);(5,6)(7,11)(8,12)(9,15)(10,14)(13,26)(16,24)(17,27)(18,28)(19,29)(20,23)(21,30)(22,32)(25,31)(33,36)(34,35)Polycyclic presentation:
2^9|a^2=ef,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302674);