#441110

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1495 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=ei,d^2=ei,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, 348306);