#501767

order	512
nilpotency class	4
isoclinism	isoclinic class of #501766
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=efh,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,f^a=fi,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408963);