#420859

order	512
nilpotency class	3
isoclinism	isoclinic class of #419041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1829 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:

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

Polycyclic presentation:

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

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, 328055);