#137127

order	512
nilpotency class	4
isoclinism	isoclinic class of #12152
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

493 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44323);