#504767

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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

610 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)(13,14,17,24,34,42,33,22,19,26,16,23,18,25,36,21)(15,27,29,39,37,43,32,20,31,38,28,44,30,41,35,40);(1,2)(3,4)(6,9)(10,11)(13,15,19,31)(14,20,26,40)(16,32,17,35)(18,37,34,30)(21,38,22,27)(23,41,24,43)(25,44,42,39)(28,36,29,33);(2,4)(14,21)(16,33)(17,36)(18,34)(20,38)(22,26)(23,42)(24,25)(27,40)(28,32)(29,35)(30,37)(39,41)(43,44);(5,7,8,12)(6,10,9,11)(14,22)(16,33)(17,36)(18,34)(20,27)(21,26)(23,25)(24,42)(28,32)(29,35)(30,37)(38,40)(39,43)(41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411963);