#499653

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

728 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)(7,10,11,12)(13,14,16,22,33,42,35,21,19,25,17,23,18,24,32,40)(15,26,28,44,36,43,34,38,27,37,29,39,30,41,31,20);(1,2)(3,4)(6,9)(10,12)(13,15,19,27)(14,20,25,38)(16,31,17,34)(18,36,33,30)(21,37,40,26)(22,41,23,43)(24,44,42,39)(28,35,29,32);(2,4)(14,21)(16,32)(17,35)(18,33)(20,37)(22,42)(23,24)(25,40)(26,38)(28,31)(29,34)(30,36)(39,41)(43,44);(5,7,8,11)(6,10,9,12)(15,27)(20,38)(26,37)(28,29)(30,36)(31,34)(39,44)(41,43)

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=efhi,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=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, 406849);