#6451815

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=gi,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=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, 6359011);