#484501

order	512
nilpotency class	4
isoclinism	isoclinic class of #484162
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
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

1025 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)(9,15)(12,20)(13,18)(14,22)(16,19)(17,24)(21,27)(23,28)(25,29)(26,31)(30,32);(2,7)(4,15)(8,18)(9,10)(11,19)(13,27)(16,28)(20,29)(21,22)(23,24)(25,32)(30,31);(3,12)(4,16)(8,20)(14,26)(15,28)(18,29)(21,30)(22,31);(1,3,5,14)(2,8,10,22)(4,13,15,27)(6,12,17,26)(7,18,9,21)(11,20,24,31)(16,25,28,32)(19,29,23,30)

Polycyclic presentation:

2^9|d^2=g,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, 391697);