#491219

order	512
nilpotency class	4
isoclinism	isoclinic class of #491113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19356	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

430 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)(5,7)(6,9)(8,12)(13,16)(14,15)(17,18)(19,28)(20,25)(21,36)(22,34)(23,27)(24,31)(26,29)(30,42)(32,44)(33,43)(35,41)(37,48)(38,45)(39,47)(40,46);(2,7)(8,15)(9,10)(11,16)(17,19,29,45,22,30,37,46,23,32,35,47,20,33,21,31)(18,24,36,43,25,39,41,44,27,40,48,42,34,38,26,28);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,18)(19,24)(20,34)(21,26)(22,25)(23,27)(28,31)(29,36)(30,39)(32,40)(33,38)(35,48)(37,41)(42,47)(43,45)(44,46);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)(17,20,23,22)(18,25,27,34)(19,30,32,33)(21,35,37,29)(24,38,40,39)(26,36,41,48)(28,42,44,43)(31,45,46,47)

Polycyclic presentation:

2^9|b^2=egh,d^2=hi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,d^b=di,d^c=di,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398415);