#481179

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	#3475	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

743 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)(9,10,12,17,14,18,24,31,13,15,23,26,25,27,36,37)(11,19,21,32,22,33,35,40,20,16,28,29,34,30,38,39);(1,2)(3,6)(4,8)(5,7)(10,15)(12,23)(16,19)(18,27)(21,28)(24,36)(30,33)(35,38);(1,3,5,4)(2,6,7,8)(9,11,13,20)(10,16,15,19)(12,21,23,28)(14,22,25,34)(17,29,26,32)(18,30,27,33)(24,35,36,38)(31,39,37,40);(11,20)(16,19)(21,28)(22,34)(29,32)(30,33)(35,38)(39,40)

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,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, 388375);