#139101

order	512
nilpotency class	4
isoclinism	isoclinic class of #139083
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	#33785	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

320 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=eh,f^c=fghi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46297);