#401624

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1772 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)(10,16)(11,21)(12,25)(13,14)(18,33)(19,20)(23,24)(26,27)(28,31)(30,38)(35,36)(39,40);(1,3)(2,5)(4,7)(6,8)(9,10,13,16,15,20,14,19)(11,17,23,28,24,32,21,31)(12,18,25,29,26,33,27,34)(22,30,36,38,37,40,35,39);(3,7)(5,8)(9,11)(10,17)(12,22)(13,23)(14,21)(15,24)(16,28)(18,30)(19,31)(20,32)(25,36)(26,37)(27,35)(29,38)(33,40)(34,39);(9,12)(10,18)(11,22)(13,25)(14,27)(15,26)(16,29)(17,30)(19,34)(20,33)(21,35)(23,36)(24,37)(28,38)(31,39)(32,40)

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 308820);