#500205

order	512
nilpotency class	4
isoclinism	isoclinic class of #499855
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#2431	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1021 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)(3,6)(4,5)(7,8)(10,16)(11,21)(12,15)(14,26)(17,32)(22,40)(23,39)(24,27)(28,38)(30,33)(34,36)(35,37);(2,5)(6,8)(9,10,13,16)(11,22,24,35)(12,20,15,18)(14,19,26,31)(17,28,30,23)(21,37,27,40)(25,36,29,34)(32,39,33,38);(1,3,4,7)(2,6,5,8)(9,11,13,24)(10,17,16,30)(12,23,15,28)(14,27,26,21)(18,22,20,35)(19,32,31,33)(25,39,29,38)(34,37,36,40);(2,5)(6,8)(9,12,13,15)(10,18,16,20)(11,23,24,28)(14,25,26,29)(17,22,30,35)(19,34,31,36)(21,38,27,39)(32,37,33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407401);