Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459299 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2752 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16690 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
949 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)(3,11,14,7)(4,15)(5,19)(6,9)(8,17)(10,21)(12,28,16,32)(13,23,31,27)(18,26)(20,25)(22,29,24,30);(1,3)(2,7)(4,16,18,12)(5,13)(6,14)(8,24,25,22)(9,11)(10,23)(15,28,26,32)(17,30,20,29)(19,27)(21,31);(1,4)(2,8)(3,12)(5,20)(6,18)(7,22)(9,25)(10,26)(11,24)(13,30)(14,16)(15,19)(17,21)(23,28)(27,32)(29,31);(8,25)(15,26)(22,24)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72251848, #72251855, #72251856, … |
| maximal normal subgroup of | #72251848, #72251855, #72251856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377793);