#440775

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
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	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1180 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)(9,10,13,26,14,17,28,18)(11,19,21,30,22,15,27,33)(12,23,24,16,25,38,40,32)(20,34,35,31,36,37,39,29);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,20)(13,27)(14,22)(16,31)(17,19)(18,33)(21,28)(23,37)(24,39)(25,36)(26,30)(29,32)(34,38)(35,40);(1,3,4,7)(2,6,5,8)(9,12,14,25)(10,16)(11,20,22,36)(13,24,28,40)(15,29)(17,32)(18,23)(19,31)(21,35,27,39)(26,38)(30,37)(33,34);(2,5)(6,8)(10,17)(13,28)(16,32)(21,27)(24,40)(30,33)(34,37)(35,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=ei,f^2=i,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,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284248, #72284369, #72284391, …
maximal normal subgroup of	#72284248, #72284369, #72284391, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 347971);