#500627

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

815 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407823);