#500566

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

858 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)(15,24)(16,27)(17,22)(18,31)(19,20)(21,29)(23,33)(25,41)(26,38)(28,43)(30,36)(32,40)(34,39)(35,44)(37,42);(1,2)(3,4)(6,9)(10,11)(13,15,17,19)(14,20,22,24)(16,25,28,34)(18,26,30,33)(21,35,37,40)(23,36,38,31)(27,39,43,41)(29,32,42,44);(2,4)(13,16,17,28)(14,21,22,37)(15,25,19,34)(18,32,30,44)(20,35,24,40)(23,39,38,41)(26,42,33,29)(27,36,43,31);(5,7,8,12)(6,10,9,11)(14,22)(20,24)(21,37)(23,38)(27,43)(31,36)(35,40)(39,41)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,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, 407762);