Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,12,8,14)(7,17,20,13,21,30,19,27)(9,23,22,32,24,15,26,31)(18,33,35,28,25,36,34,29);(6,12)(7,18)(9,24)(13,28)(17,33)(19,35)(20,34)(21,25)(22,26)(27,29)(30,36)(31,32);(2,4)(5,7)(6,13)(8,20)(9,25)(10,19)(11,21)(12,27)(14,30)(15,28)(16,17)(18,24)(22,35)(23,29)(26,34)(31,33)(32,36)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312095, #72312115, #72312138, … |
| maximal normal subgroup of | #72312095, #72312115, #72312138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54955);