#491445

order	512
nilpotency class	4
isoclinism	isoclinic class of #491439
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

577 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=f,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=di,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398641);