#441109

order	512
nilpotency class	3
isoclinism	isoclinic class of #420461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1425 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)(7,8)(9,10,13,26,14,18,28,19)(11,17,21,30,22,15,27,33)(12,23,24,16,25,38,40,32)(20,34,35,31,36,37,39,29);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,20)(13,27)(14,22)(16,31)(17,18)(19,33)(21,28)(23,37)(24,39)(25,36)(26,30)(29,32)(34,38)(35,40);(1,3,5,4)(2,7,6,8)(9,12,14,25)(10,16)(11,20,22,36)(13,24,28,40)(15,29)(17,31)(18,32)(19,23)(21,35,27,39)(26,38)(30,37)(33,34);(1,4,5,3)(2,7,6,8)(9,11)(10,17,18,15)(12,20)(13,27)(14,22)(16,31,32,29)(19,30,26,33)(21,28)(23,37,38,34)(24,39)(25,36)(35,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=ei,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348305);