#187230

order	512
nilpotency class	3
isoclinism	isoclinic class of #8479
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62752	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2040 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,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, 94426);