Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14171 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,13,6,12)(3,14,16,4,17,19,9,20)(5,21,22,10,24,26,23,27)(15,18,29,31,30,32,25,28);(1,3)(2,4)(5,15)(6,9)(7,16)(8,17)(10,18)(11,19)(12,14)(13,20)(21,28)(22,29)(23,25)(24,30)(26,31)(27,32);(1,4)(2,9)(3,11)(5,18)(6,14)(7,19)(8,20)(10,25)(12,17)(13,16)(15,26)(21,30)(22,31)(23,28)(24,32)(27,29);(1,5,7,22,8,24,6,23)(2,10,11,26,13,27,12,21)(3,15,16,29,17,30,9,25)(4,18,19,31,20,32,14,28)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213603, #72213889, #72213913, … |
| maximal normal subgroup of | #72213603, #72213889, #72213913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283060);