#503997

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×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

604 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)(16,33,30,22)(17,36,37,42)(18,26,34,44)(19,25,39,41)(21,40,38,28)(24,35,43,31);(1,2)(3,4)(6,9)(10,11)(13,15,19,31,21,17,34,16,20,32,39,35,38,37,18,30)(14,22,26,42,28,43,41,23,27,33,44,36,40,24,25,29);(2,4)(13,16,34,37,38,31,19,32,20,30,18,17,21,35,39,15)(14,23,41,24,40,42,26,33,27,29,25,43,28,36,44,22);(5,7,8,12)(6,10,9,11)(13,17,19,16,21,32,34,35,20,37,39,30,38,15,18,31)(14,24,26,29,28,22,41,42,27,43,44,23,40,33,25,36)

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^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, 411193);