#190319

order	512
nilpotency class	3
isoclinism	isoclinic class of #190208
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 97515);