#396170

order	512
nilpotency class	3
isoclinism	isoclinic class of #394127
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

1212 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=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, 303366);