#411038

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
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

759 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)(13,33)(14,21)(15,22)(16,20)(17,25)(18,31)(19,24)(26,37)(27,36)(29,40)(30,38)(32,35)(34,39);(9,11,14,26,15,27,16,25)(10,17,21,23,22,37,20,36)(12,29,31,34,32,24,30,13)(18,19,35,33,38,40,28,39);(2,5)(6,8)(9,12,15,32)(10,18)(11,24,27,29)(13,25,34,26)(14,31,16,30)(17,19)(20,28)(21,35)(22,38)(23,33)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13,14,29,15,34,16,24)(10,19,20,39,22,40,21,33)(11,12,26,31,27,32,25,30)(17,35,36,18,37,28,23,38)

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,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318234);