#482439

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

850 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,21,26,22,25,34,41)(19,23,29,37,28,36,44,42)(20,30,32,45,33,24,38,40)(27,39,43,48,31,35,46,47);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,23,25,36)(20,31)(21,29)(22,28)(24,39,30,35)(26,42,41,37)(27,33)(32,46)(34,44)(38,43)(40,47,45,48);(2,7)(6,14)(8,15)(9,16)(17,20,22,33)(18,24,25,30)(19,27,28,31)(21,32,34,38)(23,35,36,39)(26,40,41,45)(29,43,44,46)(37,47,42,48);(18,25)(21,34)(23,36)(24,30)(29,44)(32,38)(35,39)(43,46)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=hi,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fghi,f^d=fi,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, 389635);