Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×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 | |
2943 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,20,24,23)(25,26)(27,32)(28,30)(29,31);(4,8)(6,7)(10,12)(14,15)(17,19)(18,20)(21,24)(22,23)(25,27,30,29)(26,31,28,32);(2,3)(6,7)(10,12)(11,16)(18,22)(19,24)(27,29)(31,32);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,22)(21,24)(25,28,30,26)(27,32,29,31);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,14,16,15)(17,20,21,23)(18,19,22,24)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72326558, #72326599, #72326632 |
| maximal normal subgroup of | #72326558, #72326599, #72326632 |
| maximal quotient of | #72329087 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4206595);