#121487

order	512
nilpotency class	3
isoclinism	isoclinic class of #10104
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

466 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)(3,7)(4,6)(5,12)(8,11)(9,15)(10,14)(13,16)(17,18,22,26,23,27,21,38)(19,28,31,41,32,40,30,24)(20,33,36,43,37,47,35,25)(29,39,34,44,46,48,45,42);(2,6)(3,9,10,13)(7,15,14,16)(8,12)(17,19,21,30,23,32,22,31)(18,24,26,28,27,41,38,40)(20,34,35,29,37,45,36,46)(25,42,33,39,43,44,47,48);(1,3)(2,7)(4,10)(5,13)(6,14)(8,15)(9,11)(12,16)(17,20,21,35,23,37,22,36)(18,25,26,33,27,43,38,47)(19,29,31,34,32,46,30,45)(24,39,40,42,41,48,28,44)

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=e,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,f^b=fh,d^c=di,f^c=fi

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, 28683);