#489529

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	#62489	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

624 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)(19,29,33,47)(20,34,36,26)(21,25,38,44)(23,39,37,28)(24,35,43,31)(30,46,48,42)(32,41,40,45);(1,3,6,2)(4,9,12,7)(5,8,15,10)(11,14,16,13)(17,19,22,33)(18,24,27,43)(20,35,36,31)(21,30,38,48)(23,40,37,32)(25,29,44,47)(26,45,34,41)(28,46,39,42);(1,4)(2,7)(3,9)(5,14)(6,12)(8,16)(10,11)(13,15)(17,18)(19,24)(20,25)(21,34)(22,27)(23,28)(26,38)(29,35)(30,41)(31,47)(32,42)(33,43)(36,44)(37,39)(40,46)(45,48);(17,20,23,21,22,36,37,38)(18,25,28,34,27,44,39,26)(19,30,32,35,33,48,40,31)(24,41,42,29,43,45,46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,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, 396725);