Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,12,15,11,7,10,16)(8,19,22,13,23,29,21,17)(9,24,27,14,26,30,25,18)(20,31,35,33,36,28,34,32);(1,2,3,4)(5,7,11,6)(8,17)(9,24,26,30)(10,15,12,16)(13,23)(14,25,18,27)(19,22)(20,33)(21,29)(28,35)(31,34)(32,36);(2,4)(5,8)(6,13)(7,17)(9,20)(10,21)(11,23)(12,22)(14,28)(15,29)(16,19)(18,31)(24,33)(25,34)(26,36)(27,35)(30,32);(2,4)(5,9,11,26)(6,14,7,18)(8,20,23,36)(10,27,12,25)(13,28,17,31)(15,24,16,30)(19,32,29,33)(21,35,22,34)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72262977 |
| maximal normal subgroup of | #72262977 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340032);