#106990

order	512
nilpotency class	3
isoclinism	isoclinic class of #106990
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 14186);