#72334941

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

246283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(6,8)(10,12)(15,21)(16,24)(19,22)(20,25);(2,4)(5,6,7,8)(10,12)(14,18)(16,24)(19,22)(23,27);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27);(9,10)(11,12)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12);(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)

Polycyclic presentation:

2^10|b^2=j,b^a=bj,c^a=cg,c^b=cgi,d^a=dh,d^b=dhi,d^c=di,e^a=eg,e^b=eg,f^a=fi,f^b=fi

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334943
maximal normal subgroup of	#72334943
maximal quotient of	#72334970, #72334971, #72334972, …

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