#490332

order	512
nilpotency class	4
isoclinism	isoclinic class of #490087
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#466	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fh,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397528);