#491799

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19962	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

454 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309991
maximal normal subgroup of	#72309991

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398995);