Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #389036 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14533 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1271 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)(3,13,17,25)(4,18,21,10)(5,8)(6,11)(7,12)(9,16,23,14)(15,27)(19,22,20,26)(24,28)(29,32)(30,31);(2,8)(3,14)(4,19)(5,6)(9,23)(10,18)(11,12)(15,29)(16,17)(20,21)(24,32)(27,31);(1,3,6,14,7,17,5,16)(2,9,11,25,12,23,8,13)(4,15,19,28,21,29,20,30)(10,24,26,32,18,31,22,27);(1,4)(2,10)(3,15)(5,20)(6,19)(7,21)(8,22)(9,24)(11,26)(12,18)(13,27)(14,28)(16,30)(17,29)(23,31)(25,32)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237047, #72237065, #72237070, … |
| maximal normal subgroup of | #72237047, #72237065, #72237070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296232);