#409652

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

828 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316848);