Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 |
| prime radical | C2×#2410 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4071 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)(6,9)(7,11)(13,14,18,20)(15,17,16,19)(21,22)(23,27)(24,25)(26,28);(6,9)(10,12)(13,15,18,16)(14,19,20,17)(22,25)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,16)(14,19,20,17)(21,22,24,25)(23,26,28,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,17,20,19)(21,23,24,28)(22,26,25,27);(5,7)(6,10)(8,11)(9,12)(13,17)(14,16)(15,20)(18,19)(21,22)(23,27)(24,25)(26,28)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72326310, #72326314, #72326358, … |
| maximal normal subgroup of | #72326310, #72326314, #72326358, … |
| maximal quotient of | #72332942 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3768264);