#500394

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

858 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)(3,8)(4,6)(5,7)(9,10,15,19)(11,20,23,35)(12,17,14,28)(13,25,27,18)(16,26,33,22)(21,37,29,40)(24,36,30,34)(31,38,32,39);(1,3,5,4)(2,6,7,8)(11,21)(12,14)(16,31)(17,28)(20,37)(22,39)(23,29)(24,30)(26,38)(32,33)(34,36)(35,40);(1,4,5,3)(2,6,7,8)(9,11)(10,16)(12,21)(13,26)(14,29)(15,23)(17,32)(18,35)(19,33)(20,25)(22,27)(24,39)(28,31)(30,38)(34,40)(36,37);(2,7)(6,8)(9,12,15,14)(10,17,19,28)(11,21,23,29)(13,24,27,30)(16,32,33,31)(18,34,25,36)(20,37,35,40)(22,38,26,39)

Polycyclic presentation:

2^9|a^2=i,b^2=e,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, 407590);