#512422

order	512
nilpotency class	4
isoclinism	isoclinic class of #512000
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^d=ghi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419618);