Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267419 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #25585 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,18,21,24)(19,25,26,32)(20,28,30,23)(22,29,31,27);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(19,26)(22,31)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=f,b^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,f^d=fh53 conjugacy classes — view the full character table.
| maximal subgroup of | #72268339, #72268345, #72268356, … |
| maximal normal subgroup of | #72268339, #72268345, #72268356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 174625);