#484834

order	512
nilpotency class	4
isoclinism	isoclinic class of #484587
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	#40640	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

1084 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,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^c=fi,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, 392030);