#483097

order	512
nilpotency class	4
isoclinism	isoclinic class of #481466
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60718	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

739 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=be,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390293);