#401683

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1176 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=e,e^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, 308879);