Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682420 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,25,28)(23,29,30,36)(24,32,33,27)(26,34,35,31);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20);(2,4)(21,23,24,31,25,30,33,34)(22,26,27,29,28,35,32,36);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(23,30)(26,35)(29,36)(31,34);(2,4)(7,15)(10,13)(14,20)(18,19)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307483, #72307686, #72308479 |
| maximal normal subgroup of | #72307483, #72307686, #72308479 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591232);