#426543

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1030 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:

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

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=de,e^b=ei,e^c=ei,e^d=ei,g^d=gi,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, 333739);