#502386

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28225	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6,8,9)(7,11,12,10)(13,14,18,24,32,41,33,21,16,22,17,23,19,25,34,40)(15,26,30,36,43,38,42,44,28,20,29,37,31,35,27,39);(1,2)(3,4)(6,9)(10,11)(13,15,17,29,32,43,34,27,16,28,18,30,19,31,33,42)(14,20,23,36,41,35,40,44,22,26,24,37,25,38,21,39);(2,4)(14,21)(15,27)(17,33)(18,34)(19,32)(20,35)(22,40)(23,25)(24,41)(26,38)(28,42)(29,43)(30,31)(39,44);(5,7,8,12)(6,10,9,11)(14,22)(20,26)(21,40)(23,24)(25,41)(35,38)(36,37)(39,44)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409582);