#410465

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

868 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)(11,20,22,34)(12,17)(13,18)(14,28)(15,19)(16,23,30,24)(21,33,35,40)(25,31)(26,38)(27,32)(29,36,39,37);(1,3,4,7)(2,5,6,8)(9,10)(11,20)(12,17)(13,18)(14,28)(15,19)(16,24)(21,33)(22,34)(23,30)(25,31)(26,38)(27,32)(29,37)(35,40)(36,39);(9,11,14,24,15,22,13,23)(10,16,18,20,19,30,28,34)(12,21,26,37,27,35,25,36)(17,29,31,33,32,39,38,40);(2,6)(5,8)(9,12,13,25,15,27,14,26)(10,17,18,31,19,32,28,38)(11,21,23,36,22,35,24,37)(16,29,20,33,30,39,34,40)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=g,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,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, 317661);