Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,17)(8,22)(9,15)(10,16)(11,14)(12,20)(13,23)(18,26)(19,27)(21,30)(24,29)(25,28)(31,32)(33,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,26)(14,27)(15,17)(16,30)(22,35)(23,32)(24,34)(25,33)(28,36)(29,31);(2,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,28)(15,29)(16,22)(17,31)(19,32)(20,33)(21,34)(27,36)(30,35);(6,14)(12,27)(13,28)(15,16)(17,30)(22,29)(26,36)(31,35);(5,9,11,10)(6,15,14,16)(7,19,21,20)(8,23,24,25)(12,17,27,30)(13,29,28,22)(18,32,34,33)(26,31,36,35)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237505, #72237512, #72250753, … |
| maximal normal subgroup of | #72237505, #72237512, #72250753, … |
| maximal quotient of | #72306327, #72310324, #72314591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338929);