#483508

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

504 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232006, #72232010, #72232011, …
maximal normal subgroup of	#72232006, #72232010, #72232011, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390704);