#410780

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

696 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)(3,8)(4,6)(5,7)(9,10,13,17,14,19,28,18)(11,20,23,15,22,34,40,30)(12,24,25,32,27,16,26,33)(21,36,38,29,37,35,39,31);(9,11,13,23,14,22,28,40)(10,15,18,20,19,30,17,34)(12,21,26,39,27,37,25,38)(16,31,33,36,24,29,32,35);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,21)(13,26)(14,27)(15,29)(17,32)(18,33)(19,24)(20,35)(22,37)(23,39)(25,28)(30,31)(34,36)(38,40);(1,4,5,3)(2,6,7,8)(10,17,19,18)(11,22)(15,20,30,34)(16,32,24,33)(21,37)(23,40)(29,35,31,36)(38,39)

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei

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, 317976);