#205565

order	512
nilpotency class	3
isoclinism	isoclinic class of #205565
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,e^b=ei,g^b=gi,d^c=di

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, 112761);