#496804

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3002	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404000);