#491901

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	#33892	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,9,12,15)(4,13,14,7)(6,10,16,11)(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);(1,3,5,12)(2,6,8,16)(4,10,14,11)(7,15,13,9)(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);(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);(3,10)(4,14)(6,15)(7,13)(9,16)(11,12)(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,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, 399097);