Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360226 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2352 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13901 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
997 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)(4,18,22,27)(5,10)(6,11)(7,12)(8,15)(9,21,28,20)(14,29,31,23)(16,25)(17,24)(19,32,30,26);(1,3)(2,8,12,25)(4,19)(5,15)(6,16)(7,17)(9,26,28,32)(10,24,11,13)(14,20)(18,29,27,23)(21,31)(22,30);(1,4,5,20,7,22,6,21)(2,9,11,18,12,28,10,27)(3,14,15,30,17,31,16,19)(8,23,13,26,25,29,24,32);(2,10,12,11)(8,24,25,13)(9,27,28,18)(23,32,29,26)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286681, #72287308, #72287345, … |
| maximal normal subgroup of | #72286681, #72287308, #72287345, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268761);