Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456588 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16284 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1420 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)(3,13,17,14)(4,18,21,9)(5,10)(6,11)(7,12)(8,16,25,23)(15,27)(19,20,26,22)(24,30)(28,31)(29,32);(1,3,5,23,7,17,6,16)(2,8,10,14,12,25,11,13)(4,15,22,30,21,31,20,32)(9,24,26,28,18,29,19,27);(1,2)(3,14)(4,19)(5,11)(6,10)(7,12)(8,23)(9,20)(13,17)(15,29)(16,25)(18,22)(21,26)(24,31)(27,32)(28,30);(1,4)(2,9)(3,15)(5,22)(6,20)(7,21)(8,24)(10,26)(11,19)(12,18)(13,27)(14,28)(16,32)(17,31)(23,30)(25,29)Polycyclic presentation:
2^9|a^2=f,b^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237069, #72237070, #72237092, … |
| maximal normal subgroup of | #72237069, #72237070, #72237092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363784);