Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186818 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62824 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,20)(4,16)(8,19,24,31)(9,17,11,21)(14,25)(18,27)(22,32,23,28)(29,30);(1,2)(3,8)(4,17)(5,11)(6,9)(7,18)(10,13)(12,22)(14,28)(15,23)(16,20)(19,30)(21,27)(24,26)(25,31)(29,32);(3,12)(4,18)(7,17)(8,22)(14,29)(15,26)(16,27)(19,31)(20,21)(23,24)(25,30)(28,32);(1,3,13,26)(2,8,10,24)(4,14,27,29)(5,12,6,15)(7,19,20,31)(9,23,11,22)(16,25,18,30)(17,28,21,32)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260654, #72260669 |
| maximal normal subgroup of | #72260654, #72260669 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94022);