#123642

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#397
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,e^c=ei,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30838);