#151500

order	512
nilpotency class	5
isoclinism	isoclinic class of #151170
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×D₁₆
LCS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

205 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=fh,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fh,h^a=hi,d^b=df,e^b=eg,f^b=fh,h^b=hi,d^c=dg

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, 58696);