Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3671 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)(7,19)(8,25,15,28)(9,26,14,27)(10,17)(11,16)(12,18)(13,22)(20,21,30,29)(23,32)(24,31)(33,34,35,36);(5,7,11,23,12,24,10,22)(6,13,17,31,18,32,16,19)(8,20,26,34,15,30,27,36)(9,21,28,33,14,29,25,35);(2,4)(5,8)(6,14)(7,20)(9,18)(10,27)(11,26)(12,15)(13,29)(16,28)(17,25)(19,33)(21,32)(22,36)(23,34)(24,30)(31,35);(5,9)(6,15)(7,21)(8,18)(10,25)(11,28)(12,14)(13,30)(16,26)(17,27)(19,34)(20,32)(22,35)(23,33)(24,29)(31,36)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312189, #72312219, #72312265, … |
| maximal normal subgroup of | #72312189, #72312219, #72312265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299906);