Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3438 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)(5,6)(7,15)(8,12)(9,21,22,13)(10,14)(11,18,23,30)(16,27)(17,28)(19,32,33,25)(20,26)(24,31,34,36)(29,35);(5,7)(6,11)(8,17)(9,18)(10,16)(12,24)(13,15)(14,23)(19,31)(20,29)(21,27)(22,30)(25,28)(26,34)(32,35)(33,36);(2,4)(7,16)(11,23)(15,27)(17,29)(18,30)(24,34)(28,35)(31,36);(5,8,10,20)(6,12,14,26)(7,17,16,29)(9,19,22,33)(11,24,23,34)(13,25,21,32)(15,28,27,35)(18,31,30,36)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,g^b=gh,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240225, #72256595, #72280637, … |
| maximal normal subgroup of | #72240225, #72256595, #72280637, … |
| maximal quotient of | #72240229, #72281734, #72282488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400552);