#491300

order	512
nilpotency class	4
isoclinism	isoclinic class of #491233
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 ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

430 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:

(3,9)(4,11)(6,13)(8,15)(10,12)(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);(1,2)(3,6)(4,8)(5,7)(9,13)(10,14)(11,15)(12,16)(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);(1,3,5,10)(2,6,7,14)(4,12,11,9)(8,16,15,13);(2,7)(6,14)(8,15)(13,16)(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=egi,c^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=dhi,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398496);