#263570

order	512
nilpotency class	3
isoclinism	isoclinic class of #263555
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62332	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

792 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,16)(14,15)(17,18,24,28)(19,22,27,23)(20,29,35,44)(21,30,39,42)(25,34,31,33)(26,38,43,37)(32,41,36,40)(45,48,46,47);(4,13)(6,15)(7,12)(11,14)(17,19,22,28,24,27,23,18)(20,31,33,44,35,25,34,29)(21,30,37,43,39,42,38,26)(32,41,47,46,36,40,48,45);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)(17,20,24,35)(18,25,28,31)(19,29,27,44)(21,36,39,32)(22,33,23,34)(26,41,43,40)(30,46,42,45)(37,48,38,47);(3,10)(6,15)(9,16)(11,14)(17,21,23,38,24,39,22,37)(18,26,27,42,28,43,19,30)(20,32,34,48,35,36,33,47)(25,40,44,46,31,41,29,45)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gh,d^c=di,f^c=fi,g^c=gh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170766);