#481877

order	512
nilpotency class	4
isoclinism	isoclinic class of #481837
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×C₂²⋊C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

832 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,27)(20,25)(21,34,35,23)(22,26)(24,30)(28,39)(29,36,43,41)(31,38)(32,45,46,37)(33,40)(42,47,44,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(18,23,26,34)(21,35)(24,36,27,41)(25,37,40,45)(29,43)(32,46)(38,47,39,48)(42,44);(2,7)(6,14)(8,15)(9,16)(17,19,22,30)(18,24,26,27)(20,31,33,28)(21,29,35,43)(23,36,34,41)(25,39,40,38)(32,44,46,42)(37,48,45,47);(17,20,22,33)(18,25,26,40)(19,28,30,31)(21,32,35,46)(23,37,34,45)(24,38,27,39)(29,42,43,44)(36,47,41,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=hi,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,d^c=di,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, 389073);