#451757

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

911 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358953);