#112745

order	512
nilpotency class	3
isoclinism	isoclinic class of #112737
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

878 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)(17,18,22,28,23,26,21,27)(19,29,33,39,31,24,32,40)(20,25,36,42,35,43,37,44)(30,38,34,41,46,45,47,48);(1,3,6,14)(2,7,8,15)(4,13,5,12)(9,16,10,11)(17,19)(18,24)(20,34)(21,33)(22,32)(23,31)(25,41)(26,29)(27,40)(28,39)(30,36)(35,47)(37,46)(38,42)(43,48)(44,45);(2,8)(4,5)(11,16)(12,13)(17,20)(18,25)(19,30)(21,37)(22,36)(23,35)(24,38)(26,43)(27,44)(28,42)(29,45)(31,46)(32,47)(33,34)(39,48)(40,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh,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, 19941);