Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455822 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| 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 | |
853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,17,12,16,11,15)(7,18,22,30,21,9,20,13)(8,23,26,31,28,14,27,32)(19,33,24,25,35,29,34,36);(1,2,3,4)(5,7,12,21)(6,13,16,30)(8,24,28,34)(9,17,18,15)(10,22,11,20)(14,29,23,33)(19,26,35,27)(25,32,36,31);(2,4)(5,8,12,28)(6,14,16,23)(7,19,21,35)(9,29,18,33)(10,27,11,26)(13,25,30,36)(15,32,17,31)(20,24,22,34);(5,9,10,13,12,18,11,30)(6,7,17,22,16,21,15,20)(8,25,26,29,28,36,27,33)(14,24,32,35,23,34,31,19)Polycyclic presentation:
2^9|a^2=ef,b^2=fgh,c^2=i,d^2=ef,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,d^c=de,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380671);