#506410

order	512
nilpotency class	4
isoclinism	isoclinic class of #500676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

688 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,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=fh,h^a=hi,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413606);