#163875

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 71071);