#119332

order	512
nilpotency class	3
isoclinism	isoclinic class of #119328
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

564 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26528);