#110672

order	512
nilpotency class	3
isoclinism	isoclinic class of #9307
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

584 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17868);