#139651

order	512
nilpotency class	4
isoclinism	isoclinic class of #139602
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×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#23950	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46847);