#491638

order	512
nilpotency class	4
isoclinism	isoclinic class of #491572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

524 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)(6,11)(8,13)(10,15)(14,16)(17,18,22,26)(19,27,29,42)(20,32,35,38)(21,36,33,25)(23,37,39,30)(24,28,41,34)(31,48,46,44)(40,47,43,45);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15);(2,7)(6,14)(8,12)(9,15)(17,19,28,45,21,31,20,30,22,29,34,47,33,46,35,37)(18,23,38,48,25,40,24,27,26,39,32,44,36,43,41,42);(3,10)(6,14)(9,15)(11,16)(19,29)(23,39)(27,42)(30,37)(31,46)(40,43)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fghi,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=dg,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398834);