#490433

order	512
nilpotency class	4
isoclinism	isoclinic class of #490428
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

498 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)(3,7)(6,8)(9,10,13,16)(11,21,23,38)(12,20,15,18)(14,27,26,35)(17,33,31,24)(19,25,32,29)(22,39,30,36)(28,37,40,34);(1,2,3,5,4,6,7,8)(10,16)(17,31)(18,20)(19,32)(21,38)(27,35)(34,37)(36,39);(9,11)(10,17)(12,22)(13,23)(14,28)(15,30)(16,31)(18,34)(19,36)(20,37)(21,27)(24,29)(25,33)(26,40)(32,39)(35,38);(9,12,13,15)(10,18,16,20)(11,22,23,30)(14,25,26,29)(17,34,31,37)(19,35,32,27)(21,36,38,39)(24,28,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397629);