#482533

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#35692	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1094 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389729);