#484502

order	512
nilpotency class	4
isoclinism	isoclinic class of #484162
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 ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

993 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)(3,8)(4,7)(5,10)(6,11)(9,16)(12,20)(13,21)(14,18)(15,23)(17,19)(22,28)(24,27)(25,29)(26,31)(30,32);(2,7)(4,16)(8,18)(9,10)(11,19)(14,28)(17,27)(20,29)(21,24)(22,23)(25,32)(30,31);(3,12)(4,17)(8,20)(15,26)(16,27)(18,29)(22,30)(23,31);(1,3,13,26)(2,8,21,31)(4,14,27,32)(5,15,6,12)(7,18,24,30)(9,22,19,29)(10,23,11,20)(16,28,17,25)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391698);