#187131

order	512
nilpotency class	3
isoclinism	isoclinic class of #186985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62830	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2192 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:

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

Polycyclic presentation:

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

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, 94327);