#196303

order	512
nilpotency class	3
isoclinism	isoclinic class of #196299
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,f^b=fh,d^c=di,e^c=eh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103499);