#409653

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
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

818 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)(11,16,24,32,25,33,23,17)(12,26,29,36,28,37,30,18)(22,35,39,40,27,34,38,31);(2,5)(6,8)(9,11,15,23,13,25,14,24)(10,16,21,32,19,33,20,17)(12,22,30,38,28,27,29,39)(18,31,26,35,36,40,37,34);(9,11)(10,17)(12,27)(13,25)(14,24)(15,23)(16,21)(18,35)(19,32)(20,33)(22,28)(26,40)(29,38)(30,39)(31,37)(34,36);(1,3,4,7)(2,6,5,8)(9,12,15,30,13,28,14,29)(10,18,21,26,19,36,20,37)(11,22,23,38,25,27,24,39)(16,31,32,35,33,40,17,34)

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,d^c=di,e^c=ei,f^c=fi,g^c=gi

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, 316849);