#6454917

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×#3515	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3017 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=g,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^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, 6362113);