#72235720

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#95
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

28609 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 24 points:

(3,7)(5,8)(11,17)(12,20)(14,18)(15,22);(1,2)(3,8)(4,6)(5,7)(9,10)(11,18,17,14)(12,15)(13,16)(19,24,23,21)(20,22);(1,3)(2,5,6,8)(4,7)(9,11)(10,14)(12,19)(13,17)(15,21)(16,18)(20,23)(22,24);(9,12,13,20)(10,15,16,22)(11,19,17,23)(14,21,18,24);(10,16)(11,17)(15,22)(19,23)

Polycyclic presentation:

2^10|b^2=h,c^2=f,d^2=j,c^a=cij,c^b=cg,d^a=dj,e^b=ej,e^c=ej,f^b=fi,g^b=gj,g^c=gi,h^c=hj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235921, #72238075, #72251163, …
maximal normal subgroup of	#72235921, #72238075, #72251163, …
maximal quotient of	#72236751, #72263807

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, 2, F.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);