#500596

order	512
nilpotency class	4
isoclinism	isoclinic class of #500596
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13316	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

811 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)(7,10)(8,21)(9,20)(11,15)(12,17)(13,25)(14,28)(16,29)(18,23)(19,30)(22,35)(24,31)(26,33)(27,34)(32,36);(1,2)(3,4)(5,7,11,17,12,15,10,6)(8,22,24,18,27,13,26,32)(9,19,29,16,30,20,28,14)(21,23,33,35,34,36,31,25);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,26)(12,27)(14,31)(15,32)(16,33)(17,22)(19,34)(20,21)(25,29)(28,35)(30,36);(5,9,10,28,12,30,11,29)(6,14,15,20,17,16,7,19)(8,23,24,35,27,36,26,25)(13,31,32,21,22,33,18,34)

Polycyclic presentation:

2^9|b^2=h,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fh,g^c=gi,h^c=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, 407792);