#189363

order	512
nilpotency class	3
isoclinism	isoclinic class of #189298
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

879 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)(3,10,13,11,14,16,12,8)(17,18)(19,27)(20,26)(21,23)(22,25)(24,28)(29,32)(30,31);(1,3,6,13,7,14,5,12)(2,8,15,16,4,11,9,10)(18,23)(24,31)(25,26)(27,32);(1,4)(2,7)(3,11)(5,15)(6,9)(8,14)(10,13)(12,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,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, 96559);