#489105

order	512
nilpotency class	4
isoclinism	isoclinic class of #488912
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	#61251	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396301);