#105853

order	512
nilpotency class	3
isoclinism	isoclinic class of #105838
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

761 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)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,22,27,23,28,21,26)(19,29,33,39,31,24,32,40)(20,34,36,25,37,41,35,42)(30,44,47,43,45,48,46,38);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,24,28,29)(20,30)(21,32)(22,33)(23,31)(25,38,42,43)(26,39,27,40)(34,48,41,44)(35,46)(36,47)(37,45);(17,20,23,37)(18,25,28,42)(19,30,31,45)(21,36,22,35)(24,38,29,43)(26,41,27,34)(32,47,33,46)(39,44,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13049);