#418528

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1704 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)(3,4)(5,6)(7,8)(9,10,16,22)(11,19,25,32)(12,26,30,35)(13,17,24,33)(14,21,15,20)(18,28,36,29)(23,31,38,39)(27,40,37,34);(9,11,16,25)(10,17,22,33)(12,27,30,37)(13,14,24,15)(18,34,36,40)(19,20,32,21)(23,29,38,28)(26,39,35,31);(2,4)(9,12)(10,18)(11,23)(13,27)(14,28)(15,29)(16,30)(17,31)(19,34)(20,35)(21,26)(22,36)(24,37)(25,38)(32,40)(33,39);(6,8)(9,13,14,25,16,24,15,11)(10,19,20,33,22,32,21,17)(12,27,28,38,30,37,29,23)(18,34,35,39,36,40,26,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325724);