#394880

order	512
nilpotency class	3
isoclinism	isoclinic class of #394098
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302076);