#504768

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×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

590 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)(13,14,20,37,36,44,35,23,19,26,17,24,18,25,38,41)(15,27,31,39,32,43,33,21,16,30,28,40,29,42,34,22);(1,2)(3,4)(6,9)(10,11)(13,15,19,16)(14,21,26,22)(17,33,20,34)(18,32,36,29)(23,27,41,30)(24,42,37,43)(25,40,44,39)(28,38,31,35);(2,4)(13,16,19,15)(14,22,26,21)(17,34,20,33)(18,29,36,32)(23,30,41,27)(24,43,37,42)(25,39,44,40)(28,35,31,38);(5,7,8,12)(6,10,9,11)(14,23)(17,35)(18,36)(20,38)(21,27)(22,30)(24,25)(26,41)(28,33)(29,32)(31,34)(37,44)(39,43)(40,42)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=i,d^2=efh,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, 411964);