#487608

order	512
nilpotency class	4
isoclinism	isoclinic class of #486867
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	#537	C₂³
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

687 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)(19,23)(20,25)(21,24)(22,27)(26,35)(28,37)(29,36)(30,39)(31,41)(32,40)(33,43)(34,38)(42,46)(44,47)(45,48);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19,22,30)(18,23,27,39)(20,31,33,28)(21,34,35,29)(24,36,26,38)(25,41,43,37)(32,44,46,45)(40,48,42,47);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(18,24)(21,35)(23,36)(25,40)(26,27)(29,34)(32,46)(37,47)(38,39)(41,48)(42,43)(44,45);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(17,20,22,33)(18,25,27,43)(19,28,30,31)(21,32,35,46)(23,37,39,41)(24,40,26,42)(29,44,34,45)(36,47,38,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,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=dg,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, 394804);