#162549

order	512
nilpotency class	3
isoclinism	isoclinic class of #162486
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

775 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)(3,5)(4,6)(7,8)(10,15,17,21)(16,22,24,23)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(2,5)(3,8)(6,7)(9,10,14,17)(11,16,19,24)(12,15,13,21)(18,23,20,22)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(9,11,14,19)(10,16,17,24)(12,20,13,18)(15,22,21,23)(27,33)(30,36)(34,39)(37,40);(2,6)(5,7)(9,12,14,13)(10,15,17,21)(11,18,19,20)(16,23,24,22)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=i,b^a=be,d^a=dg,e^a=eg,c^b=ci,e^b=eh,f^b=fgh,d^c=dgh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 69745);