#480834

order	512
nilpotency class	4
isoclinism	isoclinic class of #480670
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

827 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,f^b=fg,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, 388030);