#501765

order	512
nilpotency class	4
isoclinism	isoclinic class of #501764
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 ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

566 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:

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

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,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408961);