#151608

order	512
nilpotency class	5
isoclinism	isoclinic class of #151512
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#2184
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

182 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58804);