#482763

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

778 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,22,28,23,27,36,43)(19,29,31,44,32,24,37,39)(20,25,33,41,34,38,46,47)(21,26,35,42,30,40,45,48);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,23,32)(18,24,27,29)(20,21,34,30)(22,31,36,37)(25,40,38,26)(28,44,43,39)(33,35,46,45)(41,42,47,48);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,23,34)(18,25,27,38)(19,30,32,21)(22,33,36,46)(24,26,29,40)(28,41,43,47)(31,45,37,35)(39,42,44,48);(3,10)(7,14)(11,16)(13,15)(17,21,23,30)(18,26,27,40)(19,20,32,34)(22,35,36,45)(24,38,29,25)(28,42,43,48)(31,33,37,46)(39,47,44,41)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,d^c=dhi,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, 389959);