#500633

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
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	#33580	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

785 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)(3,7)(4,5)(6,8)(9,10)(11,23)(12,22)(13,20)(14,18)(15,32)(16,19)(17,26)(21,30)(24,37)(25,35)(27,40)(28,36)(29,38)(31,39)(33,34);(2,5)(6,7)(9,11,13,26)(10,17,20,23)(12,25,16,33)(14,31,29,28)(15,24,30,27)(18,36,38,39)(19,35,22,34)(21,37,32,40);(10,18)(11,24)(12,16)(14,30)(15,29)(17,34)(19,32)(20,38)(21,22)(23,35)(25,31)(26,27)(28,33)(37,40);(1,3,4,8)(2,6,5,7)(9,12,13,16)(10,19,20,22)(11,25,26,33)(14,15,29,30)(17,35,23,34)(18,21,38,32)(24,28,27,31)(36,37,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407829);