#187071

order	512
nilpotency class	3
isoclinism	isoclinic class of #186976
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2005 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)(19,29,32,39,33,24,31,40)(20,34,35,42,36,25,37,41)(30,38,44,47,45,43,46,48);(1,2)(3,8)(4,7)(5,12)(6,9)(10,14)(11,16)(13,15)(17,19,23,33)(18,24,27,29)(20,30,36,45)(21,32,22,31)(25,38,34,43)(26,40,28,39)(35,46,37,44)(41,48,42,47);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,16)(12,15)(17,20,23,36)(18,25,27,34)(19,30,33,45)(21,35,22,37)(24,38,29,43)(26,41,28,42)(31,44,32,46)(39,47,40,48);(1,4,6,5)(2,7,9,12)(3,10,11,13)(8,14,16,15)(17,21,23,22)(18,26,27,28)(19,31,33,32)(20,35,36,37)(24,39,29,40)(25,41,34,42)(30,44,45,46)(38,47,43,48)

Polycyclic presentation:

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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94267);