Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493464 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1782 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62855 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2309 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)(3,11,13,24)(4,15)(5,18,20,9)(6,10)(7,14)(8,17)(12,26,27,25)(16,23,32,30)(19,22)(21,31)(28,29);(1,3)(2,7)(4,12)(5,19)(6,13)(8,21)(9,24)(10,22)(11,18)(14,20)(15,29)(16,31)(17,27)(23,26)(25,30)(28,32);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,21,22,29)(9,23,18,30)(11,25,24,26)(13,27)(14,28)(19,31)(20,32);(3,13)(7,22)(11,24)(12,27)(14,19)(21,29)(25,26)(28,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,g^b=ghi,f^c=fi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72240647, #72249653, #72258019, … |
| maximal normal subgroup of | #72240647, #72249653, #72258019, … |
| maximal quotient of | #72249667, #72262236, #72265933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400660);