#398005

order	512
nilpotency class	3
isoclinism	isoclinic class of #394353
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1233 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi,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, 305201);