#121131

order	512
nilpotency class	3
isoclinism	isoclinic class of #121129
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

338 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,e^c=eh,f^c=fhi,g^c=gi

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, 28327);