#492095

order	512
nilpotency class	4
isoclinism	isoclinic class of #491760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#12343	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

408 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399291);