#484465

order	512
nilpotency class	4
isoclinism	isoclinic class of #484157
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

1043 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)(12,23)(13,18)(14,22)(16,19)(17,25)(20,26)(21,29)(24,30)(27,32)(28,31);(2,7)(3,12)(4,15)(8,20)(9,10)(11,19)(13,28)(14,27)(16,30)(18,23)(21,32)(22,31)(24,25)(26,29);(4,16)(13,26)(15,30)(28,29);(1,3,5,14)(2,8,10,22)(4,13,15,29)(6,12,17,27)(7,18,9,21)(11,23,25,32)(16,26,30,28)(19,20,24,31)

Polycyclic presentation:

2^9|d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,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, 391661);