#6454092

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×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3077 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)(7,10)(8,9)(11,12)(13,14,19,17)(15,25,29,27)(16,31,33,22)(18,24,20,23)(21,30,36,28)(26,41,42,38)(32,44,43,37)(34,39,35,40);(2,4)(13,15,19,29)(14,21,17,36)(16,32,33,43)(18,30,20,28)(22,38,31,41)(23,27,24,25)(26,34,42,35)(37,39,44,40);(6,9)(10,12)(13,16,19,33)(14,22,17,31)(15,26,29,42)(18,35,20,34)(21,37,36,44)(23,40,24,39)(25,38,27,41)(28,32,30,43);(13,17,19,14)(15,27,29,25)(16,31,33,22)(18,23,20,24)(21,28,36,30)(26,41,42,38)(32,44,43,37)(34,39,35,40);(5,7,8,11)(6,10,9,12)(16,33)(22,31)(26,42)(32,43)(34,35)(37,44)(38,41)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,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, 6361288);