#200905

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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1087 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=hi,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, 108101);