#481177

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	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	#3480	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=gi,f^2=h,h^2=i,b^a=be,c^a=cg,e^a=eg,d^b=di,e^b=egi,f^b=fi,d^c=di

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, 388373);