#6342579

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342574
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=ci,f^a=fh,g^b=gh,e^d=eh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249775);