#203164

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
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

680 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)(3,9,14,16)(4,8,13,15)(5,12,6,11)(17,18,22,26,23,27,21,37)(19,28,32,39,30,42,31,24)(20,33,36,48,35,25,34,40)(29,43,46,38,44,41,45,47);(1,3,6,13,7,14,5,4)(2,8,12,16,10,15,11,9)(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);(1,4,5,14,7,13,6,3)(2,9,11,15,10,16,12,8)(17,20,21,34,23,35,22,36)(18,25,26,40,27,33,37,48)(19,29,31,45,30,44,32,46)(24,38,28,41,39,47,42,43);(2,10)(8,15)(9,16)(11,12)(19,30)(24,39)(28,42)(29,44)(31,32)(38,47)(41,43)(45,46)

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,f^c=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110360);