#410949

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,26)(13,19)(14,17)(15,20)(16,23)(18,28)(22,31)(24,32)(25,30)(27,34)(29,33)(35,37)(36,39)(38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16,20,32)(12,22)(13,23)(14,24)(15,25)(17,21,19,30)(18,31,34,40)(26,39,33,35)(27,36)(28,37)(29,38);(10,17,20,19)(16,30,32,21)(18,33,34,26)(31,39,40,35);(1,4,5,3)(2,6,7,8)(9,12,13,27,15,29,14,28)(10,18,17,33,20,34,19,26)(11,22,23,36,25,38,24,37)(16,31,21,35,32,40,30,39)

Polycyclic presentation:

2^9|b^2=egh,c^2=g,d^2=e,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,f^b=fi

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, 318145);