#482584

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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

877 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18,22,27,23,26,36,42)(19,28,30,43,31,24,37,40)(20,25,33,41,21,34,35,44)(29,39,45,48,32,38,46,47);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,23,31)(18,24,26,28)(20,32,21,29)(22,30,36,37)(25,39,34,38)(27,43,42,40)(33,46,35,45)(41,47,44,48);(2,8)(7,15)(9,10)(11,16)(17,20,23,21)(18,25,26,34)(19,29,31,32)(22,33,36,35)(24,38,28,39)(27,41,42,44)(30,45,37,46)(40,47,43,48);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,21,23,20)(18,25,26,34)(19,29,31,32)(22,33,36,35)(24,39,28,38)(27,44,42,41)(30,46,37,45)(40,47,43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=h,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=fgh,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, 389780);