#137006

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	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 344 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) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=h,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, 44202);