Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
3988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,19,16,18)(22,26)(24,28);(2,4)(5,6,8,9)(7,11,12,10)(13,15,17,16)(14,18,20,19)(21,22)(23,24)(25,26)(27,28);(1,2,3,4)(6,9)(10,11)(15,16)(18,19)(22,26)(23,27);(5,6)(7,11)(8,9)(10,12)(13,16,17,15)(14,19,20,18)(21,23)(22,24)(25,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,19)(16,18)(17,20)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=fh,d^2=i,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 | #72326399, #72326409, #72326443 |
| maximal normal subgroup of | #72326399, #72326409, #72326443 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3068321);