#201023

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

1030 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=h,g^2=i,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, 108219);