#140052

order	512
nilpotency class	4
isoclinism	isoclinic class of #139962
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#534	C₂×C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

419 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=g,e^2=gi,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,h^a=hi,e^b=ehi,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47248);