#72234486

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

138819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 24 points:

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

Polycyclic presentation:

2^10|b^2=j,d^2=j,e^2=j,b^a=bj,c^b=cj,d^a=dj,f^e=fh,g^e=gi,g^f=ghi

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234844, #72260698, #72260765
maximal normal subgroup of	#72234844, #72260698, #72260765
maximal quotient of	#72261558

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