#418525

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

Polycyclic presentation:

2^9|b^2=i,d^2=ei,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, 325721);