#160975

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

924 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68171);