#72225623

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

50330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 6 generators on 20 points:

(11,18)(16,20);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,20);(1,2)(3,4)(5,7,9,14)(6,10,12,17)(8,13,15,19)(11,16,18,20);(5,8)(6,11)(7,13)(9,15)(10,16)(12,18)(14,19)(17,20);(2,4);(6,12)(7,14)(11,18)(13,19)

Polycyclic presentation:

2^10|c^2=j,b^a=bg,d^a=di,e^b=eh,e^c=eh,f^b=fj,f^c=fj,g^d=gj,i^b=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225648, #72234665, #72238724, …
maximal normal subgroup of	#72225648, #72234665, #72238724, …
maximal quotient of	#72240591, #72249413, C₂×#72225623, …
automorphism group of	C₂×AΣL(1,𝔽₁₆), C₂×#352

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);