#198815

order	512
nilpotency class	3
isoclinism	isoclinic class of #198807
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=hi,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

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, 106011);