#189350

order	512
nilpotency class	3
isoclinism	isoclinic class of #189293
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

847 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,f^c=fhi,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, 96546);