#119020

order	512
nilpotency class	3
isoclinism	isoclinic class of #10496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

533 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,f^c=fi,g^c=gi

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, 26216);