#500597

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

843 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=ei,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, 407793);