Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3340 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)(5,6,10,8)(7,12,9,11)(13,14,18,20)(15,16,17,19)(21,22)(23,27)(24,26)(25,28);(5,7,10,9)(6,11,8,12)(13,15,18,17)(14,19,20,16)(21,23,26,25)(22,27,24,28);(7,9)(11,12)(14,20)(15,17)(21,22,26,24)(23,28,25,27);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,16)(14,15)(17,20)(18,19)(21,24,26,22)(23,27,25,28);(5,9,10,7)(6,11,8,12)(13,17,18,15)(14,16,20,19)(21,25,26,23)(22,27,24,28)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72262793, #72262820, #72262821, … |
| maximal normal subgroup of | #72262793, #72262820, #72262821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2005990);