#105772

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,5,6,7)(9,10,13,19,15,18,14,20)(11,21,23,33,24,16,28,31)(12,25,26,34,27,17,29,32)(22,30,36,39,38,35,37,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,23,15,24,13,28)(10,16,20,31,18,21,19,33)(12,22,26,37,27,38,29,36)(17,30,32,40,25,35,34,39);(3,8)(5,7)(9,12)(10,17)(11,22)(13,29)(14,26)(15,27)(16,30)(18,25)(19,34)(20,32)(21,35)(23,37)(24,38)(28,36)(31,40)(33,39)

Polycyclic presentation:

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

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, 12968);