#148424

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dhi,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^c=dhi,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55620);