Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| 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 | #14015 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,9)(3,13,15,8,16,22,21,24)(4,17,18,10,20,25,19,26)(14,23,29,31,30,32,28,27);(1,3,6,15,7,16,5,21)(2,8,11,22,12,24,9,13)(4,14,18,29,20,30,19,28)(10,23,25,31,26,32,17,27);(2,9,12,11)(8,22,24,13)(10,17,26,25)(23,31,32,27);(1,4)(2,10)(3,14)(5,19)(6,18)(7,20)(8,23)(9,17)(11,25)(12,26)(13,27)(15,29)(16,30)(21,28)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279719, #72317339 |
| maximal normal subgroup of | #72279719, #72317339 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275340);