#477755

order	512
nilpotency class	3
isoclinism	isoclinic class of #476112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

791 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,h^c=hi,e^d=ei,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, 384951);