#476767

order	512
nilpotency class	3
isoclinism	isoclinic class of #476144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

859 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)(9,10,15,18)(11,14,21,31)(12,22,25,19)(13,17,29,35)(16,30,33,24)(20,28,26,37)(23,36,38,34)(27,39,40,32);(1,3)(2,5)(4,7)(6,8)(9,11,14,18,15,21,31,10)(12,22,24,33,25,19,30,16)(13,26,28,17,29,20,37,35)(23,36,27,32,38,34,40,39);(2,6)(5,8)(9,12)(10,16)(11,19)(13,27)(14,30)(15,25)(17,34)(18,33)(20,32)(21,22)(23,28)(24,31)(26,39)(29,40)(35,36)(37,38);(3,7)(5,8)(9,13,14,28,15,29,31,37)(10,17,11,20,18,35,21,26)(12,23,24,27,25,38,30,40)(16,32,22,34,33,39,19,36)

Polycyclic presentation:

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

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, 383963);