Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
3367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,9,8)(7,11,12,10)(13,14)(15,17)(16,20)(18,19)(21,22)(23,27)(24,26)(25,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(14,19)(15,16)(21,23,26,25)(22,27,24,28);(2,4)(7,12)(10,11)(13,15,18,16)(14,17,19,20)(21,22,26,24)(23,28,25,27);(5,8,9,6)(7,10,12,11)(13,16,18,15)(14,17,19,20)(21,24,26,22)(23,27,25,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,17)(14,15)(16,19)(18,20)(21,25,26,23)(22,27,24,28)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=i,d^2=fg,e^2=fh,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 | #72326565, #72326596, #72326612, … |
| maximal normal subgroup of | #72326565, #72326596, #72326612, … |
| maximal quotient of | #72328614 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2148931);