#489546

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	#62484	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2965 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

639 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:

(1,2,6,3)(4,11,12,16)(5,10,15,8)(7,14,9,13)(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);(1,3,6,2)(4,9,12,7)(5,8,15,10)(11,14,16,13)(17,19,22,30)(18,23,27,41)(20,33,34,48)(21,38,35,31)(24,43,25,46)(26,47,37,42)(28,44,40,32)(29,45,39,36);(1,4)(2,7)(3,9)(5,14)(6,12)(8,16)(10,11)(13,15)(20,34)(24,25)(28,40)(29,39)(32,44)(33,48)(36,45)(43,46);(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=h,b^2=h,d^2=gi,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, 396742);