Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,12)(8,20)(9,16)(10,14)(11,15)(13,21)(17,32)(18,26)(19,28)(22,31)(23,30)(24,27)(25,33)(29,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,29)(14,26)(15,28)(16,27)(20,32)(21,35)(22,33)(23,34)(25,31)(30,36);(5,8,11,23)(6,13,15,31)(7,17,19,34)(9,22,10,21)(12,25,28,29)(14,30,16,20)(18,35,24,33)(26,32,27,36);(2,4)(6,14)(9,10)(12,26)(13,30)(15,16)(18,24)(20,31)(21,22)(25,32)(27,28)(29,36)(33,35)Polycyclic presentation:
2^9|c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72239541, #72256285, C2×#417599, … |
| maximal normal subgroup of | #72239541, #72256285, C2×#417599, … |
| maximal quotient of | #72239546, C2×#417599, #72316744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324795);