#492093

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	#33921	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

411 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)(19,29,31,24)(20,26,23,37)(22,25,35,42)(28,34,41,36)(30,44,33,39)(32,43,47,38)(40,48,45,46);(1,3,6,13)(2,7,4,10)(5,11,15,12)(8,16,14,9)(17,19,21,31)(18,24,27,29)(20,30,23,33)(22,32,35,47)(25,38,42,43)(26,39,37,44)(28,40,41,45)(34,46,36,48);(1,4,6,2)(3,10,13,7)(5,14,15,8)(9,12,16,11)(18,25)(20,23)(22,36)(24,38)(26,28)(27,42)(29,43)(30,33)(32,48)(34,35)(37,41)(39,40)(44,45)(46,47);(3,11)(5,15)(7,16)(8,14)(9,10)(12,13)(17,20,21,23)(18,26,27,37)(19,30,31,33)(22,34,35,36)(24,39,29,44)(25,41,42,28)(32,46,47,48)(38,45,43,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,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, 399289);