#441096

order	512
nilpotency class	3
isoclinism	isoclinic class of #420453
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×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1446 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)(9,10,13,16,14,18,28,19)(11,20,22,31,23,15,27,29)(12,24,25,35,26,17,33,32)(21,34,36,40,37,30,38,39);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,21)(13,27)(14,23)(16,31)(17,34)(18,20)(19,29)(22,28)(24,30)(25,38)(26,37)(32,39)(33,36)(35,40);(1,3,5,4)(2,7,6,8)(10,16,18,19)(15,29,20,31)(17,32,24,35)(30,39,34,40);(1,4,5,3)(2,7,6,8)(9,12,14,26)(10,17)(11,21,23,37)(13,25,28,33)(15,30)(16,32)(18,24)(19,35)(20,34)(22,36,27,38)(29,39)(31,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,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, 348292);