Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455690 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2825 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16284 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,13,26,11)(3,14,16,8,18,28,17,29)(4,19,21,31,22,30,15,9)(5,23,24,32,25,27,20,10);(1,3,7,18)(2,8)(4,20,22,24)(5,15,25,21)(6,16,26,17)(9,10)(11,14)(12,28)(13,29)(19,23)(27,30)(31,32);(1,4)(2,9)(3,5)(6,15)(7,22)(8,27)(10,14)(11,19)(12,30)(13,31)(16,20)(17,24)(18,25)(21,26)(23,29)(28,32);(1,5)(2,10)(3,15)(4,17)(6,20)(7,25)(8,9)(11,23)(12,27)(13,32)(14,19)(16,22)(18,21)(24,26)(28,30)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318876, #72318894, #72318910, … |
| maximal normal subgroup of | #72318876, #72318894, #72318910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363846);