Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 |
| prime radical | C2×#2410 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3843 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)(5,6,9,10)(7,8,11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,28)(25,26);(5,7)(6,8)(9,11)(10,12)(14,18)(16,19)(21,22,25,26)(23,27,28,24);(6,10)(7,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28);(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72326938, #72326973, #72327053 |
| maximal normal subgroup of | #72326938, #72326973, #72327053 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4840176);