#201071

order	512
nilpotency class	3
isoclinism	isoclinic class of #200944
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

1043 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108267);