Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,16,19,20)(22,26)(23,27);(2,3)(14,18)(16,20)(21,22)(23,24)(25,26)(27,28);(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,16,19,20)(21,23,25,27)(22,24,26,28);(2,3)(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,19)(18,20)(21,22)(23,24)(25,26)(27,28);(1,3,4,2)(5,7,8,11)(6,9,10,12)(13,16)(14,19)(15,18)(17,20)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=fg,c^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72327143, #72327157, #72327205, … |
| maximal normal subgroup of | #72327143, #72327157, #72327205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 716107);