#492413

order	512
nilpotency class	4
isoclinism	isoclinic class of #492236
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	#33908	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

403 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)(3,9,12,16)(5,14,15,8)(7,10,13,11)(17,18,23,27)(19,28,29,45)(20,25,22,37)(21,33,36,47)(24,39,40,31)(26,32,44,35)(30,46,38,43)(34,41,48,42);(1,3,6,12)(2,7,4,13)(5,10,15,11)(8,16,14,9)(19,29)(24,40)(28,45)(30,38)(31,39)(34,48)(41,42)(43,46);(1,4,6,2)(3,7,12,13)(5,14,15,8)(9,10,16,11)(17,19)(18,24)(20,30)(21,34)(22,38)(23,29)(25,41)(26,43)(27,40)(28,33)(31,35)(32,39)(36,48)(37,42)(44,46)(45,47);(3,10)(5,15)(7,16)(8,14)(9,13)(11,12)(17,20,23,22)(18,25,27,37)(19,30,29,38)(21,32,36,35)(24,41,40,42)(26,33,44,47)(28,46,45,43)(31,34,39,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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,c^b=cgi,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399609);