#487681

order	512
nilpotency class	4
isoclinism	isoclinic class of #487643
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	#18346	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

823 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)(3,7)(4,6)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,42)(29,43)(31,47)(32,39)(33,40)(34,48)(35,41)(36,44)(37,45)(38,46);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,22,29)(18,23,26,37)(20,31,32,28)(21,30,35,45)(24,38,39,36)(25,27,41,43)(33,46,34,44)(40,47,48,42);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,22,32)(18,24,26,39)(19,28,29,31)(21,34,35,33)(23,36,37,38)(25,40,41,48)(27,42,43,47)(30,46,45,44);(2,8)(3,10)(6,9)(11,16)(19,29)(20,32)(23,37)(24,39)(27,43)(30,45)(33,34)(40,48)

Polycyclic presentation:

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

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, 394877);