#106449

order	512
nilpotency class	3
isoclinism	isoclinic class of #106434
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

763 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:

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

Polycyclic presentation:

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

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, 13645);