#502942

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	#61409	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

787 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:

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

Polycyclic presentation:

2^9|a^2=e,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, 410138);