#501589

order	512
nilpotency class	4
isoclinism	isoclinic class of #501524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,e^c=ei,f^c=fh,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408785);