#72226971

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₈²	C₂⁴
commutator, socle	C₂²	C₂²×C₈²
prime radical	C₂×#41	C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

997 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(17,18)(19,24)(20,31)(21,26)(22,25)(23,27)(28,38)(29,37)(30,39)(32,36)(33,44)(34,40)(35,41)(42,47)(43,48)(45,46);(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16);(2,6)(7,12)(8,13)(14,16)(17,19,21,29,22,28,34,42,23,30,35,43,20,32,33,45)(18,24,26,37,25,38,40,47,27,39,41,48,31,36,44,46);(2,6)(7,12)(8,13)(14,16)(17,20,23,22)(18,25,27,31)(19,28,30,32)(21,33,35,34)(24,36,39,38)(26,40,41,44)(29,42,43,45)(37,46,48,47)

Polycyclic presentation:

2^10|b^2=e,c^2=f,d^2=j,e^2=g,f^2=h,g^2=i,h^2=j,c^b=ci,d^a=dj,d^b=di,d^c=dj

Conjugacy classes & character table

400 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227002
maximal normal subgroup of	#72227002

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