#398563

order	512
nilpotency class	3
isoclinism	isoclinic class of #394353
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1296 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)(3,7)(4,6)(5,8)(9,10)(11,21)(12,24)(13,27)(14,19)(15,31)(16,20)(17,23)(18,30)(22,37)(25,32)(26,33)(28,35)(29,34)(36,39)(38,40);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,15)(13,22)(14,23)(16,26)(18,32)(19,21)(20,24)(25,29)(27,40)(28,38)(30,39)(31,33)(34,36)(35,37);(1,3,5,4)(2,7,8,6)(9,12,16,11,14,26,15,23)(10,17,31,33,19,21,20,24)(13,25,30,22,28,39,29,38)(18,32,27,40,34,36,35,37);(1,4,5,3)(2,6,8,7)(9,13,14,28)(10,18,19,34)(11,22,23,38)(12,25,26,39)(15,29,16,30)(17,32,21,36)(20,35,31,27)(24,37,33,40)

Polycyclic presentation:

2^9|b^2=ei,c^2=ef,d^2=e,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, 305759);