#147921

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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

978 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)(12,25)(13,29)(14,28)(17,34)(18,20)(19,30)(21,39)(22,31)(23,33)(26,32)(27,35)(36,40)(37,38);(9,10,15,20)(11,21,24,40)(12,26,13,30)(14,32,28,19)(16,29,18,25)(17,35,37,22)(23,34,33,38)(27,36,31,39);(2,4)(9,11)(10,17)(12,27)(13,31)(14,33)(15,24)(16,34)(18,38)(19,39)(20,37)(21,30)(22,29)(23,28)(25,35)(26,40)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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^b=dh,f^b=fh,h^b=hi,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, 55117);