#504283

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

550 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)(13,14,21,28)(15,29,17,37)(16,34,32,44)(18,39,40,26)(19,41,38,25)(20,42,30,27)(22,36,24,31)(23,33,43,35);(1,2)(3,4)(6,9)(10,11)(13,15,19,32,20,33,18,31,21,17,38,16,30,35,40,36)(14,22,26,43,27,44,25,29,28,24,39,23,42,34,41,37);(2,4)(13,16,18,15,30,31,19,35,21,32,40,17,20,36,38,33)(14,23,25,22,42,29,26,34,28,43,41,24,27,37,39,44);(5,7,8,12)(6,10,9,11)(13,17,38,32,20,35,40,31,21,15,19,16,30,33,18,36)(14,24,39,43,27,34,41,29,28,22,26,23,42,44,25,37)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=fh,d^2=eh,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^b=di,d^c=di

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, 411479);