Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
3900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,8,10)(7,11,12,9)(14,18)(15,20)(21,22,25,28)(23,27,24,26);(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,19,20,16)(23,24)(26,27);(7,12)(9,11)(14,18)(16,19)(21,23,25,24)(22,26,28,27);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,18)(17,20)(21,24,25,23)(22,27,28,26);(5,7,8,12)(6,9,10,11)(13,16,17,19)(14,15,18,20)(21,22)(23,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72263345, #72263346, #72300944, … |
| maximal normal subgroup of | #72263345, #72263346, #72300944, … |
| maximal quotient of | #72330839, #72331568 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4514514);