#106045

order	512
nilpotency class	3
isoclinism	isoclinic class of #106040
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=ghi,d^b=di,e^b=ei,d^c=di,e^c=ei,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13241);