#200228

order	512
nilpotency class	3
isoclinism	isoclinic class of #200158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

982 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fi,g^c=gh

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, 107424);