Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15948 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2031 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,5,10)(3,8,14,25,15,24,20,23)(4,16,21,28,18,9,19,27)(13,29,17,26,30,22,31,32);(1,3)(2,8)(4,17)(5,20)(6,14)(7,15)(9,26)(10,23)(11,24)(12,25)(13,19)(16,32)(18,31)(21,30)(22,27)(28,29);(1,4)(2,9)(3,13)(5,21)(6,19)(7,18)(8,22)(10,27)(11,16)(12,28)(14,31)(15,30)(17,20)(23,32)(24,29)(25,26);(2,10)(5,6)(8,23)(9,27)(11,12)(14,20)(16,28)(17,31)(19,21)(22,32)(24,25)(26,29)Polycyclic presentation:
2^9|a^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216142, #72216166, … |
| maximal normal subgroup of | #72215958, #72216142, #72216166, … |
| maximal quotient of | C2×#449603, #72265722 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356799);