#493174

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

686 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)(3,6)(4,5)(7,8)(9,10)(11,23,26,29)(12,16)(13,19)(14,20)(15,30)(17,22)(18,25,35,28)(21,31)(24,36,39,40)(27,34,38,37)(32,33);(2,5)(6,8)(9,11,15,27,16,28,32,39,17,26,31,38,14,25,13,24)(10,18,21,36,12,29,19,34,22,35,30,40,20,23,33,37);(9,12)(10,14)(11,18,26,35)(13,30)(15,33)(16,22)(17,20)(19,31)(21,32)(23,28,29,25)(24,34,39,37)(27,40,38,36);(1,3,4,7)(2,6,5,8)(9,13,14,31,17,32,16,15)(10,19,20,21,22,33,12,30)(11,24,25,38,26,39,28,27)(18,34,23,36,35,37,29,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=ci,g^a=gi,c^b=ch,f^b=fi,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400370);