#505215

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

717 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)(7,10)(8,9)(11,12)(13,14,20,27)(15,23,17,32)(16,24,29,22)(18,36,38,26)(19,39,33,25)(21,40,37,28)(30,42,34,43)(31,41,35,44);(1,2)(3,4)(6,9)(10,12)(13,15,19,30,21,31,18,29,20,17,33,34,37,35,38,16)(14,22,26,41,28,42,25,23,27,24,36,44,40,43,39,32);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,31,37,34,19,15,20,29,38,35,21,30,33,17)(14,23,25,43,40,41,26,24,27,32,39,42,28,44,36,22);(6,9)(10,12)(13,17,33,30,21,35,38,29,20,15,19,34,37,31,18,16)(14,24,36,41,28,43,39,23,27,22,26,44,40,42,25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=e,d^2=fi,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

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, 412411);