Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #156684 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
2972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(6,7)(11,15)(14,16)(19,23)(21,24)(25,26,28,31)(27,30,32,29);(10,13)(11,15)(19,23)(21,24)(25,27,28,32)(26,29,31,30);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,23,24)(25,27)(26,30)(28,32)(29,31);(1,2,3,4)(5,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,19,20,23)(18,21,22,24)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=fgi,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di104 conjugacy classes — view the full character table.
| Fitting subgroup of | #72279211 |
| maximal subgroup of | #72279211, #72299670 |
| maximal normal subgroup of | #72279211, #72299670 |
| maximal quotient of | #72299905 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 65993);