#467783

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

827 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fgi,c^2=e,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,h^c=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374979);