Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #378118 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14207 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,23,28)(3,13,17,29,16,8,24,26)(4,18,20,21,10,9,22,5)(14,27,30,15,19,25,32,31);(1,3)(2,8,11,13)(4,19,10,14)(5,15)(6,17)(7,16)(9,27)(12,26,28,29)(18,25)(20,32,22,30)(21,31)(23,24);(1,4,7,10)(2,9,11,18)(3,14,16,19)(5,12,21,28)(6,22,23,20)(8,25,13,27)(15,29,31,26)(17,32,24,30);(1,5,6,18,7,21,23,9)(2,10,28,20,11,4,12,22)(3,15,17,25,16,31,24,27)(8,14,29,32,13,19,26,30)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=i,d^2=efh,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72219486, #72219491, #72219494, … |
| maximal normal subgroup of | #72219486, #72219491, #72219494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285316);