#204010

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,e^b=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111206);