#137344

order	512
nilpotency class	4
isoclinism	isoclinic class of #12155
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1889
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 80 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:

(1,2,5,9,7,10,6,15)(3,11,12,16,4,13,14,8)(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);(1,3,7,4)(2,8,10,16)(5,14,6,12)(9,13,15,11)(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);(1,4,7,3)(2,8,10,16)(5,14,6,12)(9,11,15,13)(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,c^2=g,d^2=gi,e^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=di,e^a=eghi,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44540);