#164389

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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

755 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgi

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, 71585);