#485984

order	512
nilpotency class	4
isoclinism	isoclinic class of #485605
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#3459	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

764 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,40)(30,41)(31,37)(32,38)(33,46)(34,39)(35,43)(36,45)(42,48)(44,47);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19,22,30)(18,23,26,36)(20,28,31,42)(21,29,34,45)(24,35,37,47)(25,27,39,41)(32,43,33,44)(38,40,46,48);(17,20,22,31)(18,24,26,37)(19,28,30,42)(21,33,34,32)(23,35,36,47)(25,38,39,46)(27,40,41,48)(29,44,45,43);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(20,31)(24,37)(28,42)(32,33)(35,47)(38,46)(40,48)(43,44)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393180);