#164924

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72120);