#480797

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#2404	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

865 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,d^b=di,f^b=fg,f^e=fi

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, 387993);