#440785

order	512
nilpotency class	3
isoclinism	isoclinic class of #419466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1270 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=ei,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 347981);