Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493458 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1782 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62844 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2305 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)(3,11)(4,15)(5,18,19,9)(6,10)(7,14,21,28)(8,17)(12,25)(13,24)(16,22,31,30)(20,32,29,27)(23,26);(1,3)(2,7)(4,12)(5,14)(6,13)(8,20)(9,11)(10,21)(15,29)(16,27)(17,26)(18,24)(19,28)(22,23)(25,30)(31,32);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,20,21,29)(9,22,18,30)(11,23,24,25)(13,26)(14,27)(19,31)(28,32);(3,13)(7,21)(11,24)(12,26)(14,28)(20,29)(23,25)(27,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,f^c=fi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72240647, #72249652, #72249656, … |
| maximal normal subgroup of | #72240647, #72249652, #72249656, … |
| maximal quotient of | #72249665, #72256249, #72262233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400654);