Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2403 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,9)(5,10)(6,18)(7,11)(8,14)(13,28)(15,23)(16,25)(17,27)(19,21)(20,26)(22,29)(24,30)(31,32);(1,3)(2,8)(4,13)(5,14)(6,19)(7,15)(9,24)(10,23)(11,21)(12,18)(16,29)(17,31)(20,30)(22,27)(25,28)(26,32);(8,21)(12,23)(22,24)(28,32);(1,4)(2,9)(3,13)(5,16)(6,20)(7,17)(8,22)(10,25)(11,27)(12,28)(14,29)(15,31)(18,26)(19,30)(21,24)(23,32);(1,5,7,6)(2,10,11,18)(3,14,15,19)(4,16,17,20)(8,23,21,12)(9,25,27,26)(13,29,31,30)(22,32,24,28)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216013, #72216095, #72216294, … |
| maximal normal subgroup of | #72216013, #72216095, #72216294, … |
| maximal quotient of | C2×#6437463, C4×#23897, #72249727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344659);