#112627

order	512
nilpotency class	3
isoclinism	isoclinic class of #10088
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#1438
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,e^c=eh,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, 19823);