#121353

order	512
nilpotency class	3
isoclinism	isoclinic class of #10026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

440 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=e,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^b=ei,f^b=fh,d^c=di,f^c=fh

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, 28549);