#200420

order	512
nilpotency class	3
isoclinism	isoclinic class of #200418
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35144	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,e^c=ei,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, 107616);