#485986

order	512
nilpotency class	4
isoclinism	isoclinic class of #485611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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	#23597	C₂
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

855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,23)(20,22)(21,24)(25,31)(26,28)(27,29)(30,32);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(18,22)(19,25)(23,30)(24,29)(26,32)(28,31);(19,25)(20,27)(23,31)(28,30);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,19,21,25)(18,23,24,31)(20,26,27,32)(22,28,29,30)

Polycyclic presentation:

2^9|b^2=g,d^2=gi,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=dgi,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, 393182);