Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417686 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15273 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,11)(3,14,18,9,19,28,17,29)(4,5,21,24,22,25,26,10)(15,23,20,30,32,27,31,16);(1,3)(2,9)(4,20)(5,23)(6,18)(7,17)(8,19)(10,30)(11,28)(12,14)(13,29)(15,21)(16,24)(22,31)(25,27)(26,32);(1,4)(2,10)(3,15)(5,11)(6,21)(7,26)(8,22)(9,27)(12,25)(13,24)(14,16)(17,20)(18,31)(19,32)(23,29)(28,30);(1,5,6,24,8,25,7,10)(2,4,11,21,13,22,12,26)(3,16,17,23,19,30,18,27)(9,15,14,20,29,32,28,31)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318876, #72318894, #72318969, … |
| maximal normal subgroup of | #72318876, #72318894, #72318969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327005);