#447646

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1050 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=egi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,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, 354842);