#498972

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

940 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)(3,7,8,5)(9,10)(11,20)(12,26)(13,18)(14,31)(15,19)(16,23)(17,29)(21,36)(22,37)(24,33)(25,39)(27,35)(28,34)(30,40)(32,38);(1,3,4,8)(2,5,6,7)(9,10)(11,21)(12,17)(13,31)(14,18)(15,19)(16,24)(20,36)(22,35)(23,33)(25,32)(26,29)(27,37)(28,40)(30,34)(38,39);(9,11)(10,16)(12,27)(13,25)(14,24)(15,22)(17,33)(18,35)(19,32)(20,26)(21,31)(23,29)(28,38)(30,36)(34,39)(37,40);(11,22)(16,32)(20,37)(21,35)(23,38)(24,25)(27,36)(33,39)

Polycyclic presentation:

2^9|a^2=e,b^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi,h^c=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, 406168);