#512421

order	512
nilpotency class	4
isoclinism	isoclinic class of #511998
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 ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ei,c^2=fi,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,g^d=ghi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419617);