#137329

order	512
nilpotency class	4
isoclinism	isoclinic class of #12096
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

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

normal subgroups only

This group has 448 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)(20,25,29,38)(21,26)(23,30)(24,36)(31,41)(32,39,35,48)(33,44)(34,43)(37,42)(40,47,45,46);(1,3,7,4)(2,8,10,16)(5,14,6,12)(9,13,15,11)(17,19,29,31,22,33,20,30)(18,23,38,28,27,41,25,44)(21,35,36,46,37,32,34,47)(24,39,42,45,43,48,26,40);(1,4,7,3)(2,8,10,16)(5,14,6,12)(9,11,15,13)(17,20,22,29)(18,24,27,43)(19,30,33,31)(21,36,37,34)(23,39,41,48)(25,26,38,42)(28,45,44,40)(32,47,35,46)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44525);