#398336

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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

1621 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305532);