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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1956 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)(5,6,8,10)(7,9,11,12)(14,20)(15,22)(16,18)(17,25)(21,28)(23,27);(2,3)(5,7)(6,9)(8,11)(10,12)(13,14,18,20,19,17,16,25)(15,23,24,28,22,21,26,27);(1,2,4,3)(13,15,19,22)(14,21,17,23)(16,26,18,24)(20,27,25,28);(1,3,4,2)(7,11)(9,12)(13,16,19,18)(14,20,17,25)(15,24,22,26)(21,28,23,27);(13,17,18,25,19,14,16,20)(15,23,24,28,22,21,26,27)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=g,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^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251283, #72256885, #72256888, … |
| maximal normal subgroup of | #72251283, #72256885, #72256888, … |
| maximal quotient of | C2×#7592946, #72307861 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7500142);