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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2433 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4386 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22,24,26)(23,25,28,27);(6,8)(10,12)(14,16)(18,19)(21,22)(23,27)(24,26)(25,28);(7,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,22,24,26)(23,25,28,27);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28);(5,8,9,6)(7,12,11,10)(13,16,17,14)(15,19,20,18)(21,23,24,28)(22,25,26,27)Polycyclic presentation:
2^9|a^2=gi,c^2=h,e^2=f,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 | #72232297, #72232342, #72238654, … |
| maximal normal subgroup of | #72232297, #72232342, #72238654, … |
| maximal quotient of | #72328650, #72328761, #72328763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4848879);