#425097

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1022 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)(10,16)(11,21)(13,14)(17,22)(18,31)(19,20)(23,37)(24,25)(26,34)(27,28)(35,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,10,15,20)(11,22,25,17)(12,26,29,31)(13,19,14,16)(18,28,34,27)(21,30,24,33)(23,38,39,40)(32,35,36,37);(1,4,5,2)(3,7,8,6)(9,11)(10,17)(12,23)(13,21)(14,24)(15,25)(16,30)(18,32)(19,33)(20,22)(26,38)(27,37)(28,35)(29,39)(31,40)(34,36);(3,8)(6,7)(9,12)(10,18)(11,23)(13,28)(14,27)(15,29)(16,31)(17,32)(19,26)(20,34)(21,35)(22,36)(24,37)(25,39)(30,40)(33,38)

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^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, 332293);