#6445186

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#24291	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1979 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=gi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352382);