#484910

order	512
nilpotency class	4
isoclinism	isoclinic class of #484606
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	#18298	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

763 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,23)(20,30)(21,34)(22,26)(24,33)(25,35)(27,42)(28,41)(29,38)(31,46)(32,45)(36,43)(37,44)(39,47)(40,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,29)(18,23,26,38)(20,31,32,27)(21,28,35,44)(24,39,40,36)(25,41,34,37)(30,42,45,46)(33,47,48,43);(17,20,22,32)(18,24)(19,27,29,31)(21,33,35,48)(23,36)(25,30)(26,40)(28,43,44,47)(34,45)(37,42)(38,39)(41,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(20,32)(24,40)(27,31)(30,45)(33,48)(36,39)(42,46)(43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392106);