#512420

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

576 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)(5,6)(7,19)(8,23,25,36)(9,18)(10,14)(11,26)(12,16)(13,21)(15,20,29,35)(17,22)(24,31,28,33)(27,32,34,30);(5,7,9,21,10,22,12,11)(6,13,16,19,14,26,18,17)(8,20,24,34,25,35,28,27)(15,23,30,33,29,36,32,31);(1,2,3,4)(6,14)(13,26)(15,29)(16,18)(17,19)(23,36)(30,32)(31,33);(5,8)(6,15)(7,20)(9,24)(10,25)(11,27)(12,28)(13,23)(14,29)(16,30)(17,31)(18,32)(19,33)(21,34)(22,35)(26,36)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,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, 419616);