#503015

order	512
nilpotency class	4
isoclinism	isoclinic class of #501096
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×D₄	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

813 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410211);