Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,15)(8,17)(9,13)(10,12)(11,14)(16,19)(18,20)(21,22)(23,30)(24,28)(25,35)(26,29)(27,34)(31,32)(33,36);(5,7,11,10)(6,12,14,15)(8,18,9,16)(13,20,17,19)(21,23,24,32,26,33,25,34)(22,27,35,36,29,31,28,30);(1,2,3,4)(5,8,11,9)(6,13,14,17)(7,16,10,18)(12,19,15,20)(21,22)(23,31)(24,35)(25,28)(26,29)(27,33)(30,32)(34,36);(5,9,11,8)(6,13,14,17)(7,16,10,18)(12,20,15,19)(21,24,26,25)(22,28,29,35)(23,32,33,34)(27,30,31,36);(5,10,11,7)(6,12,14,15)(8,18,9,16)(13,19,17,20)(23,33)(27,31)(30,36)(32,34)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283216, #72283258, #72283414, … |
| maximal normal subgroup of | #72283216, #72283258, #72283414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6466253);