#444394

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

904 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=g,d^2=efi,e^2=i,f^2=i,h^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,e^b=ei,f^b=fi,h^c=hi,e^d=ei,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, 351590);