#203167

order	512
nilpotency class	3
isoclinism	isoclinic class of #19420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

660 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)(3,9)(4,8)(5,12)(6,11)(7,10)(13,15)(14,16)(17,18,22,26,23,27,21,38)(19,28,32,40,30,44,31,24)(20,33,37,48,36,25,35,42)(29,45,47,39,34,43,46,41);(1,3,5,4,7,14,6,13)(2,8,11,9,10,15,12,16)(17,19)(18,24,27,40)(20,34)(21,31)(22,32)(23,30)(25,41,33,39)(26,28,38,44)(29,36)(35,47)(37,46)(42,45,48,43);(1,4,5,14,7,13,6,3)(2,9,11,15,10,16,12,8)(17,20,21,35,23,36,22,37)(18,25,26,42,27,33,38,48)(19,29,31,46,30,34,32,47)(24,39,28,43,40,41,44,45);(2,10)(8,15)(9,16)(11,12)(19,30)(24,40)(28,44)(29,34)(31,32)(39,41)(43,45)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110363);