#151655

order	512
nilpotency class	5
isoclinism	isoclinic class of #151618
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#2206
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
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

140 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=df,c^2=i,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,e^c=eg,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58851);