Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
4217 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)(5,6,8,10)(7,11,12,9)(14,18)(16,20)(21,22)(23,28)(24,27)(25,26);(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,19)(17,18)(21,23,25,27)(22,24,26,28);(7,12)(9,11)(14,18)(15,19)(22,26)(23,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,22)(23,24)(25,26)(27,28);(5,6)(7,11)(8,10)(9,12)(13,16)(14,19)(15,18)(17,20)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=fh,b^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232502, #72232503, #72232571, … |
| maximal normal subgroup of | #72232502, #72232503, #72232571, … |
| maximal quotient of | #72328399, #72329930, #72330620 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2667970);