Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 | |
4296 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)(5,6,8,9)(7,11,12,10)(13,14)(15,20)(16,17)(18,19)(21,22)(23,27)(24,25)(26,28);(6,9)(10,11)(13,15)(14,17)(16,18)(19,20)(21,23)(22,26)(24,27)(25,28);(7,12)(10,11)(14,19)(15,16)(21,23,25,28)(22,27,24,26);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,16,18,15)(14,17,19,20)(21,23)(22,26)(24,27)(25,28);(5,6)(7,11)(8,9)(10,12)(13,17)(14,16)(15,19)(18,20)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232302, #72232312, #72232356, … |
| maximal normal subgroup of | #72232302, #72232312, #72232356, … |
| maximal quotient of | #72278726, C2×#3085558, #72298175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2992754);