#120654

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27850);