#489689

order	512
nilpotency class	4
isoclinism	isoclinic class of #489398
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	#466	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

649 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)(17,18,22,27)(19,28,30,40)(20,32,34,44)(21,37,35,26)(23,39,41,29)(24,36,25,45)(31,43,38,46)(33,47,48,42);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,23)(20,33)(21,38)(22,30)(24,43)(25,46)(26,47)(27,41)(28,32)(29,36)(31,35)(34,48)(37,42)(39,45)(40,44);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(20,34)(24,25)(28,40)(29,39)(32,44)(33,48)(36,45)(43,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,35,45,22,34,21,36)(18,24,37,32,27,25,26,44)(19,29,38,48,30,39,31,33)(23,40,47,46,41,28,42,43)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=gh,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,d^c=di,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, 396885);