#404402

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1181 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 311598);