#498973

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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=i,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, 406169);