Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394046 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(8,19)(9,10)(13,26)(15,21)(16,31)(17,22)(20,23)(24,28)(25,30)(27,36)(32,33);(1,2)(3,4)(5,6)(7,13)(8,14)(9,21)(10,12)(11,15)(16,27)(17,24)(18,28)(19,30)(20,29)(22,26)(23,25)(31,35)(32,34)(33,36);(2,4)(5,7)(6,13)(8,16)(9,22)(10,17)(11,18)(12,24)(14,27)(15,28)(19,31)(20,33)(21,26)(23,32)(25,34)(29,36)(30,35);(5,8,11,20)(6,14,15,29)(7,16,18,33)(9,19,10,23)(12,25,21,30)(13,27,28,36)(17,32,22,31)(24,34,26,35)Polycyclic presentation:
2^9|d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250089, #72262403, #72262421 |
| maximal normal subgroup of | #72250089, #72262403, #72262421 |
| maximal quotient of | #72250407, #72250411, #72257693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301306);