#481176

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
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	#3444	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

779 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388372);