Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14187 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,16,8,17,24,22,14)(4,18,19,9,21,25,20,26)(15,23,30,32,31,28,29,27);(1,3)(2,8)(4,15)(5,22)(6,16)(7,17)(9,23)(10,24)(11,13)(12,14)(18,27)(19,30)(20,29)(21,31)(25,32)(26,28);(1,2)(3,14,17,8)(4,9)(5,11)(6,10)(7,12)(13,22,24,16)(15,28,31,23)(18,20)(19,25)(21,26)(27,29,32,30);(1,4,6,19,7,21,5,20)(2,9,10,25,12,26,11,18)(3,15,16,30,17,31,22,29)(8,23,24,32,14,28,13,27)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=h,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,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320134, #72342962, #72342973 |
| maximal normal subgroup of | #72320134, #72342962, #72342973 |
| maximal quotient of | #72343053 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284145);