#487629

order	512
nilpotency class	4
isoclinism	isoclinic class of #487627
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
prime radical	#18162	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1020 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)(5,9)(6,11)(10,16)(12,27)(13,20)(14,24)(15,22)(17,31)(18,21)(19,25)(23,29)(26,30)(28,32);(2,7)(4,16)(8,20)(9,10)(11,21)(13,29)(17,28)(18,30)(22,31)(23,24)(25,26)(27,32);(1,3,6,15)(2,8,11,22)(4,17,18,13)(5,14,19,12)(7,20,21,31)(9,24,25,27)(10,23,26,32)(16,28,30,29);(2,9)(3,12)(7,10)(8,22)(11,25)(13,28)(14,15)(17,29)(20,31)(21,26)(23,32)(24,27)

Polycyclic presentation:

2^9|c^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=dgi,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, 394825);