#410605

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

789 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)(3,5)(4,6)(7,8)(9,10,14,19,15,21,30,20)(11,22,24,12,25,27,17,16)(13,18,28,33,29,35,40,34)(23,36,37,26,38,39,32,31);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,21)(13,23)(14,17)(15,25)(18,31)(19,27)(20,22)(24,30)(26,35)(28,32)(29,38)(33,39)(34,36)(37,40);(9,12,15,16)(10,17)(11,20)(13,26,29,31)(14,22,30,27)(18,32)(19,25)(21,24)(23,34)(28,36,40,39)(33,38)(35,37);(2,6)(5,8)(9,13,14,28,15,29,30,40)(10,18,19,33,21,35,20,34)(11,23,17,32,25,38,24,37)(12,26,22,36,16,31,27,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317801);