#480997

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
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 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#556 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

756 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,f^b=fgi,d^c=dhi,f^e=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388193);