#160658

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#546	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh,d^c=dg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67854);