#263582

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	#3214	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

748 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)(3,7)(4,9)(5,8)(6,10)(11,14)(12,16)(13,15)(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);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,14)(10,16)(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);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,20)(18,25)(19,29)(21,36)(22,33)(23,34)(24,35)(26,41)(27,44)(28,31)(30,46)(32,39)(37,48)(38,47)(40,43)(42,45);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(17,21,22,37,24,39,23,38)(18,26,19,30,28,43,27,42)(20,32,33,47,35,36,34,48)(25,40,29,45,31,41,44,46)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ehi,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, 170778);