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 | |
8587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(13,14,20,26)(15,21,17,23)(16,24,18,22)(19,25,28,27);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,20,17)(14,21,26,23)(16,27,18,25)(19,22,28,24);(5,7)(6,9)(8,12)(10,11)(13,16,20,18)(14,22,26,24)(15,25,17,27)(19,23,28,21);(13,17,20,15)(14,23,26,21)(16,25,18,27)(19,24,28,22);(5,7)(6,10)(8,12)(9,11)(13,18,20,16)(14,24,26,22)(15,25,17,27)(19,23,28,21);(5,6)(7,9)(8,11)(10,12)(13,19,20,28)(14,25,26,27)(15,22,17,24)(16,23,18,21)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72279324, C2×#7852185, #72282610, … |
| maximal normal subgroup of | #72279324, C2×#7852185, #72282610, … |
| maximal quotient of | C2×#7852185, #72280688, #72281311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7759381);