Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2846 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
9969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14,17,18)(15,19,20,16)(22,25)(26,27);(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,26)(24,25)(27,28);(7,12)(9,11)(15,20)(16,19)(21,22,24,25)(23,27,28,26);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,26)(24,25)(27,28);(5,7)(6,9)(8,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=fg,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219164, #72228240, #72228241, … |
| maximal normal subgroup of | #72219164, #72228240, #72228241, … |
| maximal quotient of | #72210046, #72210050, #72219193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4927948);