#500460

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
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	#61287	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

779 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)(3,7)(4,5)(6,8)(9,10,13,18)(11,20,23,32)(12,26,15,17)(14,19,28,34)(16,22,33,24)(21,30,39,36)(25,35,40,38)(27,37,31,29);(2,5)(6,7)(9,10)(11,21)(12,17)(13,18)(14,29)(15,26)(16,25)(19,31)(20,36)(22,38)(23,39)(24,35)(27,34)(28,37)(30,32)(33,40);(9,11,13,23)(10,16,18,33)(12,22,15,24)(14,30,28,36)(17,32,26,20)(19,35,34,38)(21,37,39,29)(25,27,40,31);(1,3,4,8)(2,6,5,7)(9,12,13,15)(10,17,18,26)(11,22,23,24)(14,27,28,31)(16,32,33,20)(19,29,34,37)(21,38,39,35)(25,30,40,36)

Polycyclic presentation:

2^9|a^2=i,c^2=i,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,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407656);