#137852

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

295 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,6)(4,12,15,10)(5,11)(7,9)(8,14,16,13)(17,18,22,26,21,27,36,44,23,28,38,45,37,25,41,43)(19,20,30,33,29,34,40,47,31,35,42,48,46,32,24,39);(1,3,6,2,7,11,5,9)(4,10,13,8,15,12,14,16)(17,19,22,30,21,29,36,40,23,31,38,42,37,46,41,24)(18,20,26,33,27,34,44,47,28,35,45,48,25,32,43,39);(1,4,7,15)(2,8,9,16)(3,10,11,12)(5,14,6,13)(17,20)(18,24)(19,26)(21,34)(22,33)(23,35)(25,42)(27,30)(28,40)(29,44)(31,45)(32,37)(36,47)(38,48)(39,41)(43,46)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,e^b=eg,d^c=dg,e^c=egi,f^c=fg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45048);