Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #59126 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(4,8)(6,7)(9,10,13,11)(12,14,16,15)(18,21)(22,24);(1,2)(3,5)(4,6)(7,8)(17,18)(19,22)(20,21)(23,24);(2,3)(6,7)(10,11)(14,15);(9,11,13,10)(12,15,16,14);(1,3,5,2)(4,7,8,6);(1,4)(2,6)(3,7)(5,8)(9,12,13,16)(10,14,11,15)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72297795, C2×#7677402 |
| maximal normal subgroup of | #72297795, C2×#7677402 |
| maximal quotient of | #72297839, #72297898, #72298023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7584598);