Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
17565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,24)(17,20)(19,25)(21,23)(22,27)(26,28);(6,7)(13,14,17,20)(15,21,23,18)(16,19,25,24)(22,27,28,26);(10,11)(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27);(1,2,3,4)(5,7,8,6)(9,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,25)(27,28);(13,16)(14,19)(15,22)(17,25)(18,26)(20,24)(21,27)(23,28)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
| Fitting subgroup of | #72268794 |
| maximal subgroup of | #72268794, #72268797 |
| maximal normal subgroup of | #72268794, #72268797 |
| maximal quotient of | #72268824 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4837792);