#151604

order	512
nilpotency class	5
isoclinism	isoclinic class of #151520
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	C₄×C₁₆	C₂³
prime radical	#33922	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 188 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) = 256

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

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

Polycyclic presentation:

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

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, 58800);