Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470231 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(4,15)(7,18)(9,20)(11,24)(12,26)(16,29)(19,31)(22,23)(25,30)(27,32);(1,2)(3,11)(4,9)(5,8)(6,10)(7,14)(12,25)(13,24)(15,20)(16,23)(17,21)(18,28)(19,26)(22,29)(27,31)(30,32);(1,3,5,14)(2,7,8,11)(4,12,15,26)(6,13,17,28)(9,19,20,25)(10,18,21,24)(16,27,29,32)(22,31,23,30);(2,8)(4,16)(7,11)(9,22)(10,21)(12,27)(15,29)(18,24)(19,31)(20,23)(25,30)(26,32);(1,4,6,16)(2,9,10,23)(3,12,13,27)(5,15,17,29)(7,19,18,30)(8,20,21,22)(11,25,24,31)(14,26,28,32)Polycyclic presentation:
2^9|c^2=h,e^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306672, #72319784, #72340610, … |
| maximal normal subgroup of | #72306672, #72319784, #72340610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384785);