#161849

order	512
nilpotency class	3
isoclinism	isoclinic class of #161742
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

837 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:

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

Polycyclic presentation:

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

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, 69045);