#489683

order	512
nilpotency class	4
isoclinism	isoclinic class of #489390
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	#62384	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

655 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,30)(19,31,34,42)(20,35,39,46)(21,29,22,28)(24,41,44,32)(25,40,45,33)(26,38,27,37)(36,47,48,43);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,24)(20,36)(21,33)(22,40)(23,34)(25,26)(27,45)(28,43)(29,47)(30,44)(31,35)(32,38)(37,41)(39,48)(42,46);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19)(18,25)(20,32)(21,40)(22,33)(23,34)(24,26)(27,44)(28,42)(29,31)(30,45)(35,47)(36,38)(37,48)(39,41)(43,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,21,37,23,39,22,38)(18,26,28,35,30,27,29,46)(19,32,40,48,34,41,33,36)(24,42,47,45,44,31,43,25)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396879);