#6491540

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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×#3524	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2964 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)(7,11)(10,12)(13,14,17,21,19,24,18,23)(15,25,27,36,29,34,28,20)(16,22,32,37,31,39,33,40)(26,41,42,38,30,44,43,35);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,24,36)(16,30,31,26)(17,28,18,27)(21,34,23,25)(22,38,39,35)(32,42,33,43)(37,41,40,44);(6,9)(10,12)(14,21)(17,18)(20,34)(22,37)(23,24)(25,36)(27,28)(32,33)(35,44)(38,41)(39,40)(42,43);(6,9)(10,12)(13,16,19,31)(14,22,24,39)(15,26,29,30)(17,32,18,33)(20,35,36,38)(21,37,23,40)(25,41,34,44)(27,42,28,43);(5,7,8,11)(6,10,9,12)(16,31)(22,39)(26,30)(32,33)(35,38)(37,40)(41,44)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398736);