Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23930 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,27)(24,29)(25,28)(26,30)(31,34)(32,36)(33,35);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,27)(24,33)(25,31)(26,32)(28,34)(29,35)(30,36);(2,4)(22,28)(24,25)(27,34)(29,30)(31,33)(35,36);(6,11)(10,13)(12,19)(18,20);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,24,26,25)(22,29,30,28)(23,31,32,33)(27,34,36,35)Polycyclic presentation:
2^9|b^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72307685, #72307713, #72307986 |
| maximal normal subgroup of | #72307685, #72307713, #72307986 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580753);