#449784

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,g^c=gi,h^d=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, 356980);