#482626

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	#35740	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,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, 389822);