#196294

order	512
nilpotency class	3
isoclinism	isoclinic class of #196290
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

1249 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:

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

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,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, 103490);