Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455641 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
813 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17,22,34)(8,23,25,19)(9,27,31,36)(10,32,33,15)(12,21,28,20)(13,26,35,24)(14,29,18,30);(1,2,3,4)(5,7,11,22)(6,12,16,28)(8,14,25,18)(9,19,31,23)(10,21,33,20)(13,29,35,30)(15,34,32,17)(24,27,26,36);(2,4)(5,8,10,24,11,25,33,26)(6,13,15,23,16,35,32,19)(7,18,20,27,22,14,21,36)(9,28,29,34,31,12,30,17);(5,9,10,30,11,31,33,29)(6,14,15,27,16,18,32,36)(7,19,21,13,22,23,20,35)(8,17,24,12,25,34,26,28)Polycyclic presentation:
2^9|a^2=i,b^2=egh,c^2=e,d^2=f,e^2=i,f^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,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380070);