#411037

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=f,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,e^b=ei,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318233);