#6466891

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2167 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)(5,6,8,9)(7,11,12,10)(13,14)(15,20)(16,21)(17,19)(18,22)(23,34)(24,31)(25,33)(26,29)(27,36)(28,30)(32,38)(35,39)(37,43)(40,41)(42,44);(1,2,3,4)(6,9)(10,11)(14,19)(15,23)(16,27)(18,24)(20,32)(21,35)(25,40)(26,38)(28,39)(29,34)(30,36)(42,43);(5,7,8,12)(6,10,9,11)(13,15,24,38,17,26,18,23)(14,20,31,32,19,29,22,34)(16,25,39,44,28,40,27,37)(21,33,35,42,30,41,36,43);(13,16,18,27,17,28,24,39)(14,21,22,36,19,30,31,35)(15,25,23,37,26,40,38,44)(20,33,34,43,29,41,32,42);(16,28)(21,30)(25,40)(27,39)(33,41)(35,36)(37,44)(42,43)

Polycyclic presentation:

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

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, 6374087);