Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #60561 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1338 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28);(6,15)(9,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,32)(25,27)(26,31);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(19,28)(20,21)(23,27)(24,25)(29,30)(31,32);(19,29)(23,31)(27,32)(28,30)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gh,e^c=ei,f^c=fhi,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72249807, #72249808, #72249810, … |
| maximal normal subgroup of | #72249807, #72249808, #72249810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139952);