#136991

order	512
nilpotency class	4
isoclinism	isoclinic class of #136889
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

406 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=g,e^2=hi,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ehi,f^c=fh

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, 44187);