Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3890 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)(5,6,9,14,11,15,10,16)(7,17,19,28,20,27,21,12)(8,22,23,29,24,13,25,30)(18,31,32,26,33,36,34,35);(1,2,3,4)(5,7)(6,12)(8,18)(9,21)(10,19)(11,20)(13,26)(14,27)(15,28)(16,17)(22,35)(23,34)(24,33)(25,32)(29,36)(30,31);(5,8)(6,13)(7,18)(9,25)(10,23)(11,24)(12,26)(14,29)(15,22)(16,30)(17,31)(19,34)(20,33)(21,32)(27,36)(28,35);(6,14)(9,10)(12,27)(13,29)(15,16)(17,28)(19,21)(22,30)(23,25)(26,36)(31,35)(32,34)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280919, #72281084, #72281320, … |
| maximal normal subgroup of | #72280919, #72281084, #72281320, … |
| maximal quotient of | #72281619 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299921);