#146136

order	512
nilpotency class	4
isoclinism	isoclinic class of #146132
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2077
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#483
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

182 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=g,c^2=d,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fgi,c^b=cf,d^b=dg,e^b=egi,e^c=eh,f^c=fg,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, 53332);