Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
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 | |
3944 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)(7,11)(8,12)(9,10)(13,14)(15,20)(16,17)(18,19)(22,25)(23,27);(6,10)(7,12)(13,15,18,17)(14,16,19,20)(22,25)(26,28);(6,10)(8,11)(14,19)(15,17)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,26)(24,25)(27,28);(5,8,9,11)(6,7,10,12)(13,17,18,15)(14,16,19,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|b^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232286, #72232299, #72238680, … |
| maximal normal subgroup of | #72232286, #72232299, #72238680, … |
| maximal quotient of | C2×#4598429 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4505625);