Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #205795 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#390 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3180 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10)(3,11,15,12)(4,5,9,6)(8,13,16,14)(17,18)(19,22,26,29)(20,23)(21,24)(25,28,31,32)(27,30);(1,3,5,13,7,15,6,14)(2,8,4,12,10,16,9,11)(17,19)(18,22,24,29)(20,25)(21,26)(23,28,30,32)(27,31);(1,4,6,2,7,9,5,10)(3,12,14,8,15,11,13,16);(1,5,7,6)(2,9,10,4)(3,13,15,14)(8,11,16,12)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=dhi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72268390, #72268397, #72268408, … |
| maximal normal subgroup of | #72268390, #72268397, #72268408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112993);