Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(14,19)(16,24)(20,22)(23,25);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(14,19)(15,17)(16,24)(27,28);(6,8)(10,11)(13,14,18,19)(15,22,17,20)(16,21,24,26)(23,28,25,27);(5,7)(6,10)(8,11)(9,12)(13,15,18,17)(14,20,19,22)(16,23,24,25)(21,27,26,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,21)(15,23)(17,25)(18,24)(19,26)(20,27)(22,28);(5,8,9,6)(7,11,12,10)(13,17,18,15)(14,22,19,20)(16,25,24,23)(21,28,26,27)Polycyclic presentation:
2^9|c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72281661, #72282323, #72282761, … |
| maximal normal subgroup of | #72281661, #72282323, #72282761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713776);