#200831

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,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, 108027);