#151501

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	#2215	C₂²
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

208 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 46 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)(18,23)(19,27)(20,30)(22,33)(24,36)(25,39)(26,40)(28,32)(29,41)(31,46)(34,45)(35,37)(38,47)(42,43)(44,48);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18)(19,28)(20,32)(21,24)(22,35)(23,27)(25,30)(26,33)(29,44)(31,47)(34,48)(36,41)(37,46)(38,45)(39,42)(40,43);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,12,13,15)(19,29)(23,36)(27,41)(28,44)(31,42)(37,40)(39,47)(43,46)

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,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, 58697);