#6466000

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373196);