#486610

order	512
nilpotency class	4
isoclinism	isoclinic class of #486097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=dgi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393806);