#477319

order	512
nilpotency class	3
isoclinism	isoclinic class of #476144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

868 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)(3,7)(4,5)(6,8)(9,10,14,18,15,19,28,38)(11,20,23,33,24,36,30,16)(12,17,26,35,13,27,29,37)(21,32,39,34,25,40,22,31);(2,5)(6,7)(9,11)(10,16)(12,25)(13,21)(14,30)(15,24)(17,34)(18,36)(19,33)(20,38)(22,29)(23,28)(26,39)(27,31)(32,35)(37,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,26,15,13,28,29)(10,17,18,35,19,27,38,37)(11,21,23,39,24,25,30,22)(16,31,20,32,33,34,36,40);(3,8)(6,7)(9,13)(10,17)(11,22)(12,15)(14,26)(16,32)(18,37)(19,27)(20,31)(21,30)(23,25)(24,39)(28,29)(33,40)(34,36)(35,38)

Polycyclic presentation:

2^9|a^2=ef,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,g^c=gi,e^d=ei,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384515);