#200725

order	512
nilpotency class	3
isoclinism	isoclinic class of #200710
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

1002 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)(3,12,13,16)(4,8,14,15)(5,9)(6,10)(7,11)(17,18)(19,28)(20,32,33,42)(21,36)(22,26)(23,27)(24,31)(25,35,41,34)(29,44,46,38)(30,40)(37,39)(43,45,48,47);(1,3,6,15,7,13,5,8)(2,4,10,16,11,14,9,12)(17,19,21,31,23,30,22,37)(18,24,26,28,27,39,36,40)(20,29,35,47,33,46,34,45)(25,38,42,43,41,44,32,48);(1,4)(2,8)(3,10)(5,12)(6,16)(7,14)(9,13)(11,15)(17,20,23,33)(18,25,27,41)(19,29,30,46)(21,35,22,34)(24,38,39,44)(26,42,36,32)(28,43,40,48)(31,47,37,45);(1,5,7,6)(2,9,11,10)(3,8,13,15)(4,12,14,16)(19,30)(20,33)(24,39)(25,41)(28,40)(31,37)(32,42)(34,35)

Polycyclic presentation:

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

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, 107921);