#6448856

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3268 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)(7,10)(8,9)(11,12)(13,14,19,17)(15,25,29,27)(16,31,34,22)(18,23,20,24)(21,30,36,28)(26,41,43,38)(32,44,42,37)(33,40,35,39);(2,4)(13,15,19,29)(14,21,17,36)(16,32,34,42)(18,30,20,28)(22,38,31,41)(23,25,24,27)(26,33,43,35)(37,40,44,39);(6,9)(10,12)(13,16,19,34)(14,22,17,31)(15,26,29,43)(18,35,20,33)(21,37,36,44)(23,40,24,39)(25,38,27,41)(28,32,30,42);(5,7,8,11)(6,10,9,12)(13,17,19,14)(15,27,29,25)(16,31,34,22)(18,24,20,23)(21,28,36,30)(26,41,43,38)(32,44,42,37)(33,40,35,39);(13,18,19,20)(14,23,17,24)(15,28,29,30)(16,33,34,35)(21,27,36,25)(22,39,31,40)(26,42,43,32)(37,38,44,41)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356052);