#138383

order	512
nilpotency class	4
isoclinism	isoclinic class of #138195
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
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 ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 400 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=g,d^2=ghi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=ei,f^c=fi

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, 45579);