#200830

order	512
nilpotency class	3
isoclinism	isoclinic class of #200746
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

1008 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:

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

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108026);