#108749

order	512
nilpotency class	3
isoclinism	isoclinic class of #108745
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

506 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)(16,25)(20,28,24,30)(22,33,36,29)(26,37)(27,32)(31,34)(35,39,38,40);(1,2,4,7,6,8,5,3)(9,11,13,22,14,21,27,36)(10,15,18,29,17,19,32,33)(12,24,26,38,25,20,34,35)(16,30,31,40,23,28,37,39);(1,3,4,2,6,7,5,8)(9,12)(10,16,17,23)(11,20)(13,26)(14,25)(15,28,19,30)(18,31,32,37)(21,24)(22,35)(27,34)(29,39,33,40)(36,38)

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,c^b=ch,f^b=fhi,e^c=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15945);