#374289

order	512
nilpotency class	3
isoclinism	isoclinic class of #196110
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1502 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)(4,5)(6,8)(9,10,15,20,13,18,14,19)(11,21,24,36,23,16,29,31)(12,17,28,35,26,33,27,34)(22,37,25,40,38,30,39,32);(2,5)(6,7)(9,11)(10,16)(12,25)(13,23)(14,29)(15,24)(17,32)(18,21)(19,36)(20,31)(22,27)(26,39)(28,38)(30,34)(33,40)(35,37);(9,12,15,27,13,26,14,28)(10,17,19,35,18,33,20,34)(11,22,24,39,23,38,29,25)(16,30,36,32,21,37,31,40);(1,3,4,8)(2,6,5,7)(10,18)(16,21)(17,33)(19,20)(30,37)(31,36)(32,40)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281485);