#162324

order	512
nilpotency class	3
isoclinism	isoclinic class of #162118
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

838 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 69520);