Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| 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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(14,15)(18,20);(5,7)(6,9)(8,12)(10,11)(14,18)(15,20)(17,19);(1,2,3,4)(13,14,17,18,16,15,19,20);(6,10)(9,11);(7,12)(9,11)(13,15,17,20,16,14,19,18)Polycyclic presentation:
2^9|c^2=ghi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72256932, #72256972, #72257099 |
| maximal normal subgroup of | #72256932, #72256972, #72257099 |
| maximal quotient of | #72265016 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721637);