#487620

order	512
nilpotency class	4
isoclinism	isoclinic class of #487619
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
prime radical	#12090	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

975 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)(3,12)(4,7)(5,10)(6,11)(8,13)(9,16)(14,21)(15,25)(17,20)(18,24)(19,26)(22,27)(23,30)(28,31)(29,32);(2,7)(4,16)(8,19)(9,10)(11,20)(12,21)(14,26)(17,30)(22,31)(23,24)(25,32)(28,29);(3,13)(4,17)(8,12)(14,28)(15,27)(16,30)(19,21)(22,25)(26,29)(31,32);(1,3,6,15)(2,8,11,22)(4,14,17,29)(5,13,18,27)(7,19,20,31)(9,21,23,32)(10,12,24,25)(16,26,30,28)

Polycyclic presentation:

2^9|d^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=dg,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, 394816);