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 | |
6105 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)(3,15)(4,18)(5,11)(6,12)(7,13)(8,14)(9,16)(10,19)(17,26)(20,30)(21,29)(22,31)(23,32)(24,28)(25,27);(1,3)(2,9)(4,5)(6,25)(7,16)(8,17)(10,28)(11,30)(12,15)(13,26)(14,27)(18,32)(19,23)(20,22)(21,24)(29,31);(1,4)(2,10)(3,5)(6,19)(7,20)(8,21)(9,11)(12,29)(13,18)(14,30)(15,32)(16,22)(17,24)(23,25)(26,31)(27,28);(1,5)(2,11)(3,4)(6,22)(7,23)(8,24)(9,10)(12,31)(13,32)(14,28)(15,18)(16,19)(17,21)(20,25)(26,29)(27,30);(1,6,8,7)(2,12,14,13)(3,16,17,25)(4,19,21,20)(5,22,24,23)(9,26,27,15)(10,29,30,18)(11,31,28,32)Polycyclic presentation:
2^9|e^2=i,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 | #72213603, #72214082, #72215901, … |
| maximal normal subgroup of | #72213603, #72214082, #72215901, … |
| maximal quotient of | #72235210, #72286816, #72287110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344783);