#483640

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3171	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=dg,e^a=eg,d^b=di,e^b=eg,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320856, #72321994, #72322062
maximal normal subgroup of	#72320856, #72321994, #72322062

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390836);