Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(8,16)(11,17)(15,20)(18,19)(21,22,27,24,26,30,25,29)(23,31,34,33,32,28,35,36);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,24)(23,32)(25,27)(28,33)(29,30)(31,36);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23)(22,28)(24,36)(25,35)(26,32)(27,34)(29,33)(30,31);(7,14)(10,13)(15,20)(18,19)(21,24,25,22,26,29,27,30)(23,33,35,31,32,36,34,28);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,26,27)(22,24,30,29)(23,34,32,35)(28,33,31,36)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307631, #72307818, #72307838, … |
| maximal normal subgroup of | #72307631, #72307818, #72307838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088457);