#106534

order	512
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

765 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:

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

Polycyclic presentation:

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

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, 13730);