Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437585 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2403 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6079 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)(3,13)(4,10)(5,11)(6,9)(7,12)(8,16)(14,28)(15,23)(17,24)(18,27)(19,22)(20,26)(21,31)(25,30)(29,32);(1,3)(2,8)(4,14)(5,19)(6,16)(7,15)(9,23)(10,25)(11,13)(12,22)(17,31)(18,29)(20,30)(21,27)(24,28)(26,32);(2,9)(5,6)(8,13)(10,24)(11,12)(16,19)(17,20)(21,28)(22,23)(25,32)(26,27)(30,31);(1,4)(2,10)(3,14)(5,20)(6,17)(7,18)(8,21)(9,24)(11,26)(12,27)(13,28)(15,29)(16,31)(19,30)(22,25)(23,32);(3,15)(8,22)(13,23)(14,29)(16,19)(21,25)(28,32)(30,31)Polycyclic presentation:
2^9|f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,f^c=fi,g^c=gi,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215901, #72216141, #72219683, … |
| maximal normal subgroup of | #72215901, #72216141, #72219683, … |
| maximal quotient of | #72219483, #72236660, #72251192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344781);