#459489

order	512
nilpotency class	3
isoclinism	isoclinic class of #459488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1043 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366685);