#120830

order	512
nilpotency class	3
isoclinism	isoclinic class of #10560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

395 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28026);