#502943

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

727 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=f,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410139);