#487643

order	512
nilpotency class	4
isoclinism	isoclinic class of #487643
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
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,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, 394839);