#199577

order	512
nilpotency class	3
isoclinism	isoclinic class of #199522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3213	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1183 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106773);