#501525

order	512
nilpotency class	4
isoclinism	isoclinic class of #501524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

637 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=ehi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,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, 408721);