#482877

order	512
nilpotency class	4
isoclinism	isoclinic class of #482734
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×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#556 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

755 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fghi,d^c=dh,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, 390073);