Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2527 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,12)(9,11)(13,14)(15,20)(16,21)(17,26)(18,22)(19,23)(24,27)(25,28);(2,3)(5,7,10,6)(8,12,9,11);(1,2,4,3)(13,15)(14,19)(16,23)(17,25)(18,24)(20,26)(21,27)(22,28);(5,8,10,9)(6,11,7,12)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,28)(22,27);(1,3,4,2)(5,9,10,8)(6,11,7,12)(13,16,18,17)(14,21,22,26)(15,23,24,25)(19,27,28,20)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
| maximal subgroup of | #72251224, #72256864, #72256935 |
| maximal normal subgroup of | #72251224, #72256864, #72256935 |
| maximal quotient of | #72265663 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718828);