#144570

order	512
nilpotency class	4
isoclinism	isoclinic class of #144569
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2023
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 275 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)(3,10,12,16)(4,8)(5,14)(6,9)(7,13,15,11)(17,18)(19,27,29,43)(20,32)(21,25)(22,37)(23,30,28,45)(24,35)(26,36)(31,40,38,44)(33,42,46,39)(34,47)(41,48);(3,11,12,13)(7,10,15,16)(17,18)(19,28,29,23)(20,24)(21,25)(22,26)(27,30,43,45)(31,42,38,39)(32,35)(33,44,46,40)(34,41)(36,37)(47,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,16)(9,15)(10,14)(17,19,21,29)(18,23,25,28)(20,33,34,46)(22,38,36,31)(24,40,41,44)(26,42,37,39)(27,32,43,47)(30,35,45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51766);