#108748

order	512
nilpotency class	3
isoclinism	isoclinic class of #108744
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1434
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

526 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15944);