#492118

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3391	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

407 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=de,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399314);