Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245615 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #47830 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12)(3,7,13,9)(4,8,16,10)(6,11,18,27)(14,21,28,25)(15,22,29,20)(17,23,32,26)(19,31,24,30);(2,7)(3,13)(8,22)(9,12)(10,20)(11,21)(14,28)(15,29)(19,23)(24,26)(25,27)(30,31);(2,8)(3,14)(4,16)(7,19)(9,24)(10,12)(11,23)(13,28)(15,31)(17,32)(20,25)(21,22)(26,27)(29,30);(2,7)(3,13)(4,17)(8,19)(9,12)(10,24)(11,21)(14,28)(15,31)(16,32)(20,26)(22,23)(25,27)(29,30)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fh,e^d=eh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72213216 |
| maximal normal subgroup of | #72213216 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152812);