Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394482 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2435 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14724 |
| prime radical | #63090 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,7,14,6,13)(3,15,18,9,20,29,19,28)(4,10,27,30,22,31,21,17)(5,23,26,32,25,16,24,11);(1,3)(2,9)(4,11)(5,17)(6,19)(7,20)(8,18)(10,26)(12,29)(13,15)(14,28)(16,21)(22,32)(23,27)(24,31)(25,30);(1,4)(2,10)(3,16)(5,9)(6,27)(7,22)(8,21)(11,19)(12,17)(13,30)(14,31)(15,26)(18,32)(20,23)(24,29)(25,28);(1,5)(2,11)(3,17)(4,9)(6,26)(7,25)(8,24)(10,19)(12,16)(13,23)(14,32)(15,27)(18,31)(20,30)(21,29)(22,28)Polycyclic presentation:
2^9|a^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213603, #72213913, #72219482, … |
| maximal normal subgroup of | #72213603, #72213913, #72219482, … |
| maximal quotient of | #72262966 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302608);