#511880

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1116 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:

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

Polycyclic presentation:

2^9|b^2=ei,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419076);