#200216

order	512
nilpotency class	3
isoclinism	isoclinic class of #200146
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	#60962	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1029 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107412);