#186388

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62788	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2183 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93584);