Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417630 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15327 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,5,10,7,12)(3,13,16,25,15,8,22,24)(4,17,19,27,21,28,20,9)(14,26,18,23,31,32,30,29);(1,3)(2,8)(4,18)(5,15)(6,16)(7,22)(9,26)(10,13)(11,24)(12,25)(14,20)(17,23)(19,31)(21,30)(27,32)(28,29);(1,4,7,19,5,21,6,20)(2,9,12,17,10,27,11,28)(3,14,22,18,15,31,16,30)(8,23,25,32,13,29,24,26);(1,2)(3,13)(4,9)(5,10)(6,11)(7,12)(8,15)(14,29)(16,25)(17,19)(18,26)(20,28)(21,27)(22,24)(23,31)(30,32)Polycyclic presentation:
2^9|a^2=f,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72280350, #72287261, #72287308, … |
| maximal normal subgroup of | #72280350, #72287261, #72287308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332208);