Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418818 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15494 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
989 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)(3,15)(4,21,17,30)(5,25,16,32)(6,13)(7,12)(8,14)(9,18)(10,24,31,23)(11,22,29,26)(19,28)(20,27);(1,3)(2,9)(4,22)(5,24)(6,19)(7,18)(8,20)(10,29)(11,31)(12,27)(13,15)(14,28)(16,23)(17,26)(21,25)(30,32);(1,4)(2,10,14,31)(3,16)(5,20)(6,23)(7,24)(8,17)(9,25,28,32)(11,15,29,27)(12,21,13,30)(18,22)(19,26);(1,5)(2,11)(3,17)(4,20)(6,22)(7,26)(8,16)(9,21)(10,15)(12,32)(13,25)(14,29)(18,23)(19,24)(27,31)(28,30)Polycyclic presentation:
2^9|a^2=eg,c^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338514, #72338518, #72338528 |
| maximal normal subgroup of | #72338514, #72338518, #72338528 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341170);