Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1806 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)(5,6,11,16)(7,17,21,29)(8,14,24,28)(9,13,10,15)(12,20,26,18)(19,33,35,31)(22,30,23,27)(25,32,36,34);(5,7)(6,12)(8,19)(9,20)(10,18)(11,21)(13,17)(14,25)(15,29)(16,26)(22,32)(23,34)(24,35)(27,31)(28,36)(30,33);(2,4)(6,13)(7,18)(8,22)(9,10)(14,28)(15,16)(17,29)(20,21)(23,24)(25,33)(31,36)(32,34);(5,8,11,24)(6,14,16,28)(7,19,21,35)(9,23,10,22)(12,25,26,36)(13,27,15,30)(17,31,29,33)(18,32,20,34)Polycyclic presentation:
2^9|a^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361887);