#484523

order	512
nilpotency class	4
isoclinism	isoclinic class of #484181
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

765 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)(21,33)(22,26)(24,32)(25,35)(27,42)(28,40)(29,38)(31,41)(34,39)(36,44)(37,46)(43,47)(45,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,29)(18,23,26,38)(20,27,31,43)(21,28,35,46)(24,36,39,48)(25,40,33,37)(30,47,41,42)(32,44,34,45);(17,20,22,31)(18,24,26,39)(19,27,29,43)(21,34,35,32)(23,36,38,48)(25,41,33,30)(28,45,46,44)(37,47,40,42);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(20,31)(24,39)(27,43)(30,41)(32,34)(36,48)(42,47)(44,45)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,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, 391719);