#491220

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

424 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)(17,18,22,27)(19,28,32,44)(20,25,23,34)(21,36,37,48)(24,38,40,31)(26,35,42,29)(30,43,33,45)(39,47,41,46);(2,7)(8,15)(9,10)(11,16)(17,19,29,46,23,30,37,38,22,32,35,47,20,33,21,31)(18,24,36,45,25,41,42,44,27,40,48,43,34,39,26,28);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,18)(19,24)(20,34)(21,26)(22,27)(23,25)(28,31)(29,36)(30,41)(32,40)(33,39)(35,48)(37,42)(38,44)(43,47)(45,46);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)(17,20,22,23)(18,25,27,34)(19,30,32,33)(21,35,37,29)(24,39,40,41)(26,36,42,48)(28,43,44,45)(31,46,38,47)

Polycyclic presentation:

2^9|a^2=i,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, 398416);