Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,8)(7,10,12,11)(14,19)(16,24)(17,21)(20,26)(23,27)(25,28);(2,3)(5,7)(6,10)(8,11)(9,12)(13,14,17,19,18,16,21,24)(15,20,23,26,22,25,27,28);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,15)(14,20)(16,25)(17,23)(18,22)(19,26)(21,27)(24,28);(1,3,4,2)(7,12)(10,11)(14,16)(15,22)(19,24)(23,27);(13,16,17,24,18,14,21,19)(15,20,23,26,22,25,27,28)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=hi,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251228, #72251252, #72256885, … |
| maximal normal subgroup of | #72251228, #72251252, #72256885, … |
| maximal quotient of | #72264564, C2×#7604432 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511628);