#136201

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gh,d^2=gi,e^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,h^a=hi,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=egi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43397);