#503993

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

591 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)(7,11)(8,9)(10,12)(13,14,20,27)(15,29,32,23)(16,22,34,43)(17,36,33,42)(18,39,30,26)(19,40,38,25)(21,41,37,28)(24,35,44,31);(1,2)(3,4)(6,9)(10,11)(13,15,19,31,21,33,18,16,20,32,38,35,37,17,30,34)(14,22,26,42,28,24,25,29,27,43,39,36,41,44,40,23);(2,4)(13,16,18,17,37,31,19,32,20,34,30,33,21,35,38,15)(14,23,25,24,41,36,26,22,27,29,40,44,28,42,39,43);(5,7,8,12)(6,10,9,11)(13,17,38,16,21,15,30,35,20,33,19,34,37,32,18,31)(14,24,39,23,28,43,40,42,27,44,26,29,41,22,25,36)

Polycyclic presentation:

2^9|a^2=i,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^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, 411189);