#483585

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

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	#3169	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

503 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cg,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	#72286610, #72286643, #72286672, …
maximal normal subgroup of	#72286610, #72286643, #72286672, …
maximal quotient of	#72286672, #72286988, #72287125, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390781);