#200925

order	512
nilpotency class	3
isoclinism	isoclinic class of #200920
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	#60688	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1031 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)(3,12,13,16)(4,8,14,15)(5,9)(6,10)(7,11)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,31)(25,36)(30,44)(32,40)(33,39)(35,41)(37,42)(38,47)(43,46)(45,48);(1,3,6,15,7,13,5,8)(2,4,10,16,11,14,9,12)(17,19,21,31,23,33,22,32)(18,24,27,29,28,40,26,39)(20,30,36,46,37,47,35,45)(25,38,34,43,41,44,42,48);(1,4)(2,8)(3,10)(5,12)(6,16)(7,14)(9,13)(11,15)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,42)(27,34)(28,41)(29,43)(31,46)(32,45)(33,47)(39,48)(40,44);(1,5,7,6)(2,9,11,10)(3,8,13,15)(4,12,14,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,39,40,29)(25,34,41,42)(30,45,47,46)(38,43,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=h,b^a=be,c^a=cf,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108121);