Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
3967 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,10)(7,8)(14,17)(18,19)(22,25)(23,28);(5,6)(7,11)(8,12)(9,10)(13,14,16,17)(15,19,20,18)(21,22)(23,27)(24,25)(26,28);(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,18,17,19)(21,23,24,28)(22,26,25,27);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(5,6)(7,12)(8,11)(9,10)(13,15)(14,19)(16,20)(17,18)(21,23)(22,27)(24,28)(25,26)Polycyclic presentation:
2^9|b^2=fg,c^2=f,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72326299, #72326304, #72326331, … |
| maximal normal subgroup of | #72326299, #72326304, #72326331, … |
| maximal quotient of | #72328472, #72329113, #72330351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3248183);