#501654

order	512
nilpotency class	4
isoclinism	isoclinic class of #501652
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fh,h^c=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, 408850);