#489600

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62341	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

677 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

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, 396796);