#430547

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

874 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:

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

Polycyclic presentation:

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

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, 337743);