#72234412

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃₂² · Schur multiplier (H₂) = C₄²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈²	C₄²
commutator	C₄	C₃₂²
Frattini, cosocle	C₄×C₁₆²	C₂²
socle, prime radical	C₂³	#92895
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈² ◁ G
LECS	G ▷ C₄×C₁₆² ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

1129 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 1129 conjugacy classes of subgroups — too many to draw. Only its 651 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 4096

A minimal faithful permutation representation — 2 generators on 80 points:

(1,2)(3,9,10,15)(4,8)(5,7)(6,11,13,16)(12,14)(17,18,19,23,20,24,27,33,21,25,28,34,30,36,39,43,22,26,29,35,31,37,40,44,32,38,41,45,42,46,47,48);(1,3)(2,6,7,13)(4,11)(5,10)(8,9,14,15)(12,16)(49,50,51,55,52,56,59,65,53,57,60,66,62,68,71,75,54,58,61,67,63,69,72,76,64,70,73,77,74,78,79,80)

Polycyclic presentation:

2^12|a^2=c,b^2=d,c^2=f,d^2=g,e^2=h,f^2=i,g^2=j,i^2=k,j^2=l,b^a=be,c^b=ch,d^a=dh

Conjugacy classes & character table

1408 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

#72234417

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", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);