#108569

order	512
nilpotency class	3
isoclinism	isoclinic class of #108562
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1434
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,f^b=fi,e^c=eh

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, 15765);