Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394328 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2617 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14725 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2154 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)(3,7)(4,14,15,8)(5,17)(6,10)(9,19)(11,24)(12,21,26,27)(13,22)(16,23,29,28)(18,20)(25,32,30,31);(3,11,13,18)(7,20,22,24)(12,25,26,30)(21,31,27,32);(1,3)(2,7)(4,12)(5,18)(6,13)(8,21)(9,24)(10,22)(11,19)(14,27)(15,26)(16,30)(17,20)(23,32)(25,29)(28,31);(1,4)(2,8)(3,12)(5,16)(6,15)(7,21)(9,23)(10,14)(11,25)(13,26)(17,28)(18,30)(19,29)(20,31)(22,27)(24,32)Polycyclic presentation:
2^9|a^2=f,b^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285925, #72286050, #72286194, … |
| maximal normal subgroup of | #72285925, #72286050, #72286194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302723);