Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2668 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2758 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)(4,16)(5,10)(6,20)(7,11)(8,14)(9,18)(13,30)(15,25)(17,26)(19,27)(21,23)(22,28)(24,32)(29,31);(1,3)(2,8)(4,17)(5,14)(6,21)(7,15)(9,26)(10,25)(11,23)(12,20)(13,18)(16,24)(19,31)(22,32)(27,30)(28,29);(8,23)(12,25)(24,30)(26,29);(1,4)(2,9)(3,13)(5,18)(6,22)(7,19)(8,24)(10,27)(11,28)(12,29)(14,31)(15,32)(16,20)(17,21)(23,30)(25,26);(1,5,7,6)(2,10,11,20)(3,14,15,21)(4,18,19,22)(8,25,23,12)(9,27,28,16)(13,31,32,17)(24,26,30,29)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215721, #72216047, #72216095, … |
| maximal normal subgroup of | #72215721, #72216047, #72216095, … |
| maximal quotient of | #72250795, C2×#7500747, #72256170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407943);