#462444

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 369640);