#110250

order	512
nilpotency class	3
isoclinism	isoclinic class of #110242
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

571 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=ehi,f^c=fhi

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, 17446);