#136978

order	512
nilpotency class	4
isoclinism	isoclinic class of #136906
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

337 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=gh,d^2=g,e^2=h,f^2=gi,g^2=i,h^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,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi,h^c=hi

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, 44174);