Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456654 |
| 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 | |
1469 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)(3,7,12,20)(4,14)(5,17)(6,9)(8,16)(10,18)(11,25,26,32)(13,29,28,22)(15,24)(19,23)(21,30,31,27);(1,3)(2,7,9,20)(4,11)(5,13)(6,12)(8,21,23,31)(10,22,17,29)(14,25,24,32)(15,26)(16,27)(18,28)(19,30);(5,18)(7,20)(8,23)(10,17)(11,26)(13,28)(16,19)(21,31);(1,4)(2,8)(3,11)(5,19)(6,15)(7,21)(9,23)(10,24)(12,26)(13,30)(14,17)(16,18)(20,31)(22,32)(25,29)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285398, #72286235, #72286242, … |
| maximal normal subgroup of | #72285398, #72286235, #72286242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363850);