#477320

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

Polycyclic presentation:

2^9|a^2=efi,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, 384516);