#499531

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
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×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

838 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efhi,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,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, 406727);