#6466866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2172 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374062);