#491154

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	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

401 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(18,24)(19,30)(20,22)(21,34)(25,35)(26,41)(27,38)(28,37)(29,39)(31,46)(32,33)(36,43)(42,48)(44,47);(17,18,25,38,22,27,21,24,23,29,34,37,20,28,35,39)(19,26,40,47,30,43,31,36,32,44,45,41,33,42,46,48);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,32)(18,26,29,44)(20,33,22,30)(21,31,35,46)(24,36,39,48)(25,40,34,45)(27,43,28,42)(37,41,38,47);(2,7)(6,15)(8,12)(9,16)(17,20,23,22)(18,27,29,28)(19,30,32,33)(21,25,35,34)(24,37,39,38)(26,42,44,43)(31,45,46,40)(36,47,48,41)

Polycyclic presentation:

2^9|b^2=egi,c^2=h,d^2=i,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, 398350);