Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806551 |
| 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 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,22,27,31)(23,32,34,30)(24,29,26,35)(25,28,33,36);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,23,26,25,27,34,24,33)(22,28,29,32,31,36,35,30);(6,12)(10,13)(11,18)(17,19)(21,24,27,26)(22,29,31,35)(23,33,34,25)(28,32,36,30);(7,15)(10,13)(14,20)(17,19)(21,25,26,34,27,33,24,23)(22,30,29,28,31,32,35,36)Polycyclic presentation:
2^9|b^2=i,c^2=fh,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256954, #72257047, #72257071, … |
| maximal normal subgroup of | #72256954, #72257047, #72257071, … |
| maximal quotient of | #72264917 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714405);