Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,11,14,25)(8,19)(9,22)(12,24,28,35)(13,23,26,15)(16,30)(17,33)(20,21)(27,34,36,29)(31,32);(5,6)(7,12)(8,13)(9,23)(10,14)(11,20)(15,22)(16,27)(17,34)(18,28)(19,25)(21,26)(24,31)(29,33)(30,35)(32,36);(2,4)(5,7,10,18)(6,12,14,28)(8,16,21,32)(9,17,22,33)(11,24,25,35)(13,27,26,36)(15,29,23,34)(19,30,20,31);(6,13)(8,20)(9,22)(11,23)(12,27)(14,26)(15,25)(16,31)(17,33)(19,21)(24,34)(28,36)(29,35)(30,32)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72249996, #72250863, #72250867, … |
| maximal normal subgroup of | #72249996, #72250863, #72250867, … |
| maximal quotient of | #72287160, #72287201, #72309426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404620);