Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| 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 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,24)(15,20,26,35)(16,28,31,40)(17,23,18,22)(21,33,38,30)(25,41,29,44)(27,39,32,36)(34,42,37,43);(1,2,3,4)(7,8)(10,11)(13,15)(14,20)(16,29)(17,32)(18,27)(19,26)(21,37)(22,39)(23,36)(24,35)(25,31)(28,44)(30,42)(33,43)(34,38)(40,41);(5,7,9,8)(6,10,12,11)(13,16,19,31)(14,21,24,38)(15,25,26,29)(17,30,18,33)(20,34,35,37)(22,28,23,40)(27,42,32,43)(36,44,39,41);(5,8,9,7)(6,11,12,10)(15,26)(20,35)(25,29)(27,32)(34,37)(36,39)(41,44)(42,43);(6,12)(10,11)(13,17,19,18)(14,22,24,23)(15,27,26,32)(16,30,31,33)(20,36,35,39)(21,28,38,40)(25,42,29,43)(34,44,37,41)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394906);