#502485

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28484	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

720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=i,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,g^c=gh,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, 409681);