#203173

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
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

654 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)(3,11)(4,5)(6,10)(7,9)(8,13)(12,15)(14,16)(17,18,22,26,23,27,21,37)(19,28,32,39,30,42,31,24)(20,33,36,48,34,25,35,40)(29,43,46,38,45,41,44,47);(1,3,5,13,7,15,6,14)(2,8,10,11,9,16,4,12)(17,19)(18,24,27,39)(20,29)(21,31)(22,32)(23,30)(25,38,33,47)(26,28,37,42)(34,45)(35,44)(36,46)(40,41,48,43);(1,4,5,2,7,10,6,9)(3,12,13,8,15,11,14,16)(17,20,21,35,23,34,22,36)(18,25,26,40,27,33,37,48)(19,29,31,44,30,45,32,46)(24,38,28,41,39,47,42,43);(2,9)(4,10)(8,16)(11,12)(19,30)(20,34)(24,39)(25,33)(28,42)(31,32)(35,36)(40,48)

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,d^b=dhi,d^c=dh,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, 110369);