#466884

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

725 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374080);