#144604

order	512
nilpotency class	4
isoclinism	isoclinic class of #144603
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2023
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

307 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,9,11,15)(4,7)(5,8)(6,12,10,13)(14,16)(17,18)(19,28,30,44)(20,33)(21,26)(22,27)(23,38)(24,31,40,46)(25,37)(29,45,32,42)(34,48)(35,43,47,39)(36,41);(1,2)(3,10,11,6)(4,7)(5,8)(9,12,15,13)(14,16)(19,29,30,32)(20,34)(24,39,40,43)(25,41)(28,42,44,45)(31,47,46,35)(33,48)(36,37);(1,3)(2,6)(4,13)(5,11)(7,15)(8,10)(9,16)(12,14)(17,19,22,30)(18,24,27,40)(20,35,34,47)(21,32,23,29)(25,42,41,45)(26,43,38,39)(28,33,44,48)(31,37,46,36)

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=i,e^2=ghi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,e^b=ei,f^b=fgi,d^c=dg,e^c=egh,f^c=fi,h^c=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51800);