#134609

order	512
nilpotency class	4
isoclinism	isoclinic class of #124718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

357 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=i,d^2=i,e^2=h,f^2=gh,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41805);