#487614

order	512
nilpotency class	4
isoclinism	isoclinic class of #486875
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×C₄⋊C₄	C₂⁴
prime radical	#12156	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

738 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18)(19,27)(20,24)(21,34)(22,26)(23,30)(25,36)(28,43)(29,42)(31,39)(32,46)(33,48)(35,38)(37,44)(40,47)(41,45);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(20,31)(24,39)(25,34)(27,42)(28,32)(33,45)(37,40)(44,47);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,19,22,29)(18,23,26,38)(20,32,31,28)(21,35,36,30)(24,40,39,37)(25,42,34,27)(33,44,45,47)(41,43,48,46);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(17,20,22,31)(18,24,26,39)(19,28,29,32)(21,33,36,45)(23,37,38,40)(25,41,34,48)(27,43,42,46)(30,44,35,47)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394810);