#489240

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	#61401	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

719 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:

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

Polycyclic presentation:

2^9|a^2=h,c^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, 396436);