Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418473 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2673 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15412 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1587 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)(10,19)(11,26,27,32)(13,30,29,22)(15,25)(18,24)(21,31,23,28);(1,3)(2,7,9,20)(4,11,15,27)(5,13)(6,12)(8,23)(10,22,17,30)(14,32)(16,28,18,31)(19,29)(21,24)(25,26);(7,20)(21,23)(22,30)(26,32);(1,4)(2,8)(3,11,12,27)(5,18)(6,15)(7,21,20,23)(9,24)(10,25)(13,31,29,28)(14,17)(16,19)(22,32,30,26)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72215972, #72216077, … |
| maximal normal subgroup of | #72215958, #72215972, #72216077, … |
| maximal quotient of | #72237667, #72251301, C2×#430055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337251);