#151745

order	512
nilpotency class	5
isoclinism	isoclinic class of #151734
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#2246	C₂²
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

167 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=di,c^2=eg,d^2=f,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,h^a=hi,c^b=cf

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58941);