#481109

order	512
nilpotency class	3
isoclinism	isoclinic class of #17547
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

771 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388305);